Definite Integrals · MathLopedia

Problem 1

Easy Basic Definite Integrals

Find $\int_0^4 x \, dx$ using the limit of Riemann sums (divide $[0,4]$ into $n$ equal parts and take $\xi_i = x_i$), and interpret the number geometrically.

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Problem 2

Easy Basic Definite Integrals

Calculate $\int_0^1 e^x \, dx$ using the limit of Riemann sums.

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Problem 3

Medium Basic Definite Integrals

Write $\int_0^\pi \sin x \, dx$ in the form of an infinite series (Riemann sum limit) and find the limit of that series.

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Problem 4

Easy Basic Definite Integrals

Calculate $\int_0^{10} a^x \, dx$ by forming the Riemann sum.

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Problem 5

Medium Basic Definite Integrals

Find using the Riemann-sum definition: a) $\int_a^b \frac{1}{x} \, dx$; b) $\int_a^b \cos x \, dx$.

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Problem 6

Easy Basic Definite Integrals

Using the Newton-Leibniz formula, calculate $\int_{-1}^1\left(1-x^2\right) \, dx$.

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Problem 7

Easy Basic Definite Integrals

Using basic properties of definite integrals, prove that $1^{\circ}\ \int_{-1}^1 \arcsin x \, dx=0$; $2^{\circ}\ \int_{-a}^a \sinh x \, dx=0$; $3^{\circ}\ \int_{-\frac{\pi}{4}}^{\frac{\pi}{4}} \tan x \, dx=0$.

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Problem 8

Easy Basic Definite Integrals

Discuss the geometric interpretation of integrating odd functions over symmetric intervals, then apply the Newton-Leibniz formula to calculate $\int_0^1 x^n \, dx$.

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Problem 9

Easy Basic Definite Integrals

Discuss the geometric interpretation of $\int_0^1 x^n \, dx = \frac{1}{n+1}$, then apply the Newton-Leibniz formula to calculate $\int_0^\pi \sin^2 x \, dx$.

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Problem 10

Easy Basic Definite Integrals

Discuss the geometric interpretation of $\int_0^\pi \sin^2 x \, dx = \frac{\pi}{2}$, then apply the Newton-Leibniz formula to calculate $\int_0^a \frac{dx}{\sqrt{a^2-x^2}}$.

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Problem 11

Easy Basic Definite Integrals

Apply the Newton-Leibniz formula to calculate the following integral: $\int_0^1 \frac{dx}{1+x^2}$.

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Problem 12

Easy Basic Definite Integrals

Apply the Newton-Leibniz formula to calculate the following integral: $\int_1^2 \frac{1}{x^x} \, dx$.

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Problem 13

Medium Basic Definite Integrals

Apply the Newton-Leibniz formula to calculate the following integral: $\int_0^1 \sinh x \, dx$.

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Problem 14

Easy Basic Definite Integrals

Using the definite integral, find the limit of the sum: $\lim_{n \rightarrow \infty}\left(\frac{1}{n}+\frac{1}{n+1}+\cdots+\frac{1}{2n}\right)$.

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Problem 15

Medium Basic Definite Integrals

Using the definite integral, find the limit of the sum: $\lim_{n \rightarrow \infty}\left(\frac{1}{n^2}+\frac{2}{n^2}+\cdots+\frac{2n-1}{n^2}\right)$.

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Problem 16

Easy Basic Definite Integrals

Using the definite integral, find the limit of the sum: $\lim_{n \rightarrow \infty}\left(\frac{1^p}{n^{p+1}}+\frac{2^p}{n^{p+1}}+\cdots+\frac{n^p}{n^{p+1}}\right)$, for $p>0$.

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Problem 17

Easy Basic Definite Integrals

Calculate the integral: $\int_0^8(1+\sqrt{2x}+\sqrt[3]{x}) \, dx$.

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Problem 18

Easy Basic Definite Integrals

Calculate the integral: $\int_{-\frac{\pi}{4}}^0 \frac{3x^4+3x^2+1}{x^2+1} \, dx$.

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Problem 19

Easy Basic Definite Integrals

Calculate the integral: $\int_{-\frac{\pi}{4}}^{\frac{\pi}{4}} \frac{dx}{\cos^2 x}$.

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Problem 20

Medium Basic Definite Integrals

Calculate the integral: $\int_1^2 x(\ln x+1) \, dx$.

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Problem 21

Easy Basic Definite Integrals

Calculate the integral: $\int_{-\pi}^\pi x(\sin x-1) \, dx$.

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Problem 22

Easy Basic Definite Integrals

Calculate the integral: $\int_{-\pi}^\pi x(2+\cos x) \cos^{17} x \, dx$.

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Problem 23

Easy Basic Definite Integrals

Calculate the integral: $\int_1^4 \frac{1+\sqrt{x}}{x^2} \, dx$.

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Problem 24

Easy Basic Definite Integrals

Calculate the integral: $\int_0^1 \sqrt[m]{x^n} \, dx$, where $m, n \in \mathbb{N}$.

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Problem 25

Medium Basic Definite Integrals

Calculate the integral: $\int_1^e \frac{\sin(\ln x)}{x} \, dx$.

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Problem 26

Easy Basic Definite Integrals

Calculate the integral: $\int_0^{\frac{T}{2}} \sin\left(\frac{2\pi t}{T}-\varphi_0\right) dt$.

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Problem 27

Easy Basic Definite Integrals

Calculate the integral: $\int_{-\frac{\sqrt{2}}{2}}^{\frac{\sqrt{2}}{2}} \frac{dx}{\sqrt{1-x^2}}$.

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Problem 28

Easy Basic Definite Integrals

Calculate the integral: $\int_{-\frac{1}{2}}^{\frac{1}{2}} \frac{dx}{1-x^2}$.

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Problem 29

Easy Basic Definite Integrals

Calculate the integral: $\int_0^1 \frac{dx}{e^x+e^{-x}}$.

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Problem 30

Easy Basic Definite Integrals

Calculate the integral: $\int_0^{\ln 2} \frac{e^x-1}{\sqrt{e^x-1}} \, dx$.

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Problem 31

Medium Basic Definite Integrals

Calculate the integral: $\int_e^{e^e} x^3 e^{x-4} \, dx$.

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Problem 32

Easy Basic Definite Integrals

Calculate the integral: $\int_0^{\frac{\pi}{2}} \sin x(1+\cos^2 x) \, dx$.

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Problem 33

Easy Basic Definite Integrals

Calculate the integral: $\int_0^{\frac{\pi}{2}} \cos x \sin^2 x \, dx$.

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Problem 34

Easy Basic Definite Integrals

Calculate the integral: $\int_0^R \sqrt{R^2-x^2} \, dx$.

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Problem 35

Easy Basic Definite Integrals

Calculate the integral: $\int_0^R x^2 \sqrt{R^2-x^2} \, dx$.

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Problem 36

Easy Basic Definite Integrals

Calculate the integral: $\int_0^4 x \sqrt{x^2+9} \, dx$.

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Problem 37

Medium Basic Definite Integrals

Calculate the integral: $\int_0^4 x^3 \sqrt{x^2+9} \, dx$.

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Problem 38

Medium Basic Definite Integrals

Calculate the integral: $\int_0^{\ln 5} \frac{\sqrt{e^x-1}}{1+3e^{-x}} \, dx$.

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Problem 39

Medium Basic Definite Integrals

Calculate the integral: $\int_5^0 \frac{dx}{2x+\sqrt{3x+1}}$.

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Problem 40

Easy Basic Definite Integrals

Calculate the integral: $\int_{\frac{1}{\pi}}^{\frac{2}{\pi}} \frac{1}{x^2} \sin \frac{1}{x} \, dx$.

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Problem 41

Easy Substitution and Symmetry

Calculate the integral: $\int_0^1 \frac{x^3}{x^6+2x^3+1} \, dx$.

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Problem 42

Easy Substitution and Symmetry

Calculate the integral: $\int_0^1 \frac{x^3-2x+3}{\sqrt{x^2+1}} \, dx$.

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Problem 43

Easy Substitution and Symmetry

Transform $\int_a^b f(x) \, dx$ using a linear substitution into an integral with given limits $\alpha$ and $\beta$.

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Problem 44

Easy Substitution and Symmetry

Prove the following equality: $\int_a^b f(a+b-x) \, dx = \int_a^b f(x) \, dx$.

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Problem 45

Easy Substitution and Symmetry

Prove the following equality: $\int_0^{\pi/2} f(\sin x) \, dx = \int_0^{\pi/2} f(\cos x) \, dx$.

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Problem 46

Easy Substitution and Symmetry

1) Prove the equality $\int_0^\pi x f(\sin x) \, dx = \frac{\pi}{2} \int_0^\pi f(\sin x) \, dx$, where $f$ is continuous on $[0, \pi]$. 2) Calculate $\int_0^\pi x \frac{\sin x}{1+\cos^2 x} \, dx$.

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Problem 47

Easy Substitution and Symmetry

Calculate the integral: $\int_{\frac{\pi}{4}}^{\frac{\pi}{3}} \frac{dx}{\cos^2 x(1+\tan x)^2}$.

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Problem 48

Easy Substitution and Symmetry

Calculate the integral: $\int_0^{2\pi} \frac{(a+\cos x) \, dx}{1+2a\cos x+a^2}$.

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Problem 49

Easy Substitution and Symmetry

Prove: 1) $\int_0^{\pi/2} \sin^n x \, dx = \int_0^{\pi/2} \cos^n x \, dx$. 2) $\int_0^{\pi/2} \frac{\sqrt{\sin^{2k} x + \cos^{2k} x}}{\cos 2x} \, dx = 0$.

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Problem 50

Easy Substitution and Symmetry

Find the value of $x$ for which $\int_1^x \frac{dt}{t\sqrt{t^2-1}} = \frac{\pi}{3}$.

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Problem 51

Easy Substitution and Symmetry

If $u(x)$ and $v(x)$ are differentiable on $[a, b]$, prove that $\int_a^b u(x) v'(x) \, dx = \left.u(x) v(x)\right|_a^b - \int_a^b v(x) u'(x) \, dx$ (Integration by Parts for Definite Integrals).

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Problem 52

Easy Substitution and Symmetry

Calculate the integral: $\int_0^1 \arcsin x \, dx$.

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Problem 53

Easy Substitution and Symmetry

Calculate the integral: $\int_0^{\pi/2} e^x \sin x \, dx$.

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Problem 54

Medium Substitution and Symmetry

Prove that $\int_0^{\pi/2} \sin^n x \, dx = \frac{n-1}{n} \int_0^{\pi/2} \sin^{n-2} x \, dx$, and use this to find $\int_0^{\pi/2} \sin^6 x \, dx$.

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Problem 55

Easy Substitution and Symmetry

Prove: 1) $\int_0^{\pi/2} \sin^{2n} x \, dx = \frac{(2n-1)!!}{(2n)!!} \cdot \frac{\pi}{2}$. 2) $\int_0^{\pi/2} \cos^{2k} t \, dt = \frac{(2k-1)!!}{(2k)!!} \cdot \frac{\pi}{2}$.

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Problem 56

Easy Substitution and Symmetry

Prove the following equality: $\int_{-1}^1 (1-x^2)^n \, dx = \frac{2^{2n+1} \cdot (n!)^2}{(2n+1)!}$.

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Problem 57

Easy Substitution and Symmetry

Let $I(a, b) = \int_a^b \frac{(1-x^2) \, dx}{(1+x^2)\sqrt{1+x^4}}$. 1) Show that (i) $I(a, b) = I(-b, -a)$; (ii) $I(a, b) = I(1/a, 1/b)$ ($ab > 0$); (iii) $I(a, 1/a) = 0$. 2) Calculate $I(a, b)$ (first for $a>1,b>1$, using $t=x+1/x$, then use the results from part 1).

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Problem 58

Easy Substitution and Symmetry

Let $n$ and $m$ be natural numbers. Prove that $\int_0^\pi \left(2\cos\frac{x}{2}\right)^n \cos\left(\frac{n}{2}-m\right)x \, dx = \pi\binom{n}{m}$.

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Problem 59

Easy Substitution and Symmetry

Calculate the integral: $\int_0^1 x^n e^{-x} \, dx$.

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Problem 60

Easy Substitution and Symmetry

Calculate the integral: $\int_1^e x^m(\ln x)^n \, dx$, where $m$ and $n$ are non-negative integers.

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Problem 61

Medium Substitution and Symmetry

Calculate $\int \frac{x^2 \, dy}{x^{5/3}+y^{5/3}}$ and $\int \frac{y^2 \, dx}{x^{5/3}+y^{5/3}}$ if $x = a\cos^3 t$, $y = a\sin^3 t$; $t \in [0, \frac{\pi}{2}]$.

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Problem 62

Easy Substitution and Symmetry

Consider $\int_0^3 \frac{dx}{(x-2)^2}$. Obviously $\int \frac{dx}{(x-2)^2} = -\frac{1}{x-2}$, so $\int_0^3 \frac{dx}{(x-2)^2} = -\left.\frac{1}{x-2}\right|_0^3 = -\frac{3}{2}$. But the integrand $\frac{1}{(x-2)^2}$ is positive in the given interval, so the integral over the given limits must be positive. Where is the error?

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Problem 63

Easy Substitution and Symmetry

The integral $J = \int_0^\pi \sqrt{\frac{1+\cos 2x}{2}} \, dx$ is calculated as follows: $$\int_0^\pi \sqrt{\frac{1+\cos 2x}{2}} \, dx = \int_0^\pi \sqrt{\cos^2 x} \, dx = \int_0^\pi \cos x \, dx = \left.\sin x\right|_0^\pi = 0.$$ However, the integrand $f(x) = \sqrt{\frac{1+\cos 2x}{2}}$ is positive, so the value of the integral must be greater than zero. Where is the error?

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Problem 64

Easy Substitution and Symmetry

If we apply the substitution $t = \tan \frac{x}{2}$ to the integral $\int_0^{2\pi} \frac{dx}{4-3\cos x}$, we get: $$ \int_0^{2\pi} \frac{dx}{4-3\cos x} = \int_0^0 \frac{2dt}{(1+t^2)\left(4-3\cdot\frac{1-t^2}{1+t^2}\right)} = 0 $$ However, the integrand $f(x) = \frac{1}{4-3\cos x}$ is non-negative! Where is the error?

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Problem 65

Medium Substitution and Symmetry

If the function $f(x)$ satisfies the conditions of the Newton-Leibniz theorem (Fundamental Theorem of Calculus) for $x \in [a, b]$, prove that: $$ \frac{d}{dx}\left(\int_a^x f(t) \, dt\right) = f(x) $$

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Problem 66

Hard Substitution and Symmetry

Find the maximum and minimum values of the function: $$ f(x) = \int_0^x \frac{2t+1}{t^2-2t+2} \, dt \quad \text{for} \quad x \in [1, \sqrt{3}+1] $$

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Problem 67

Hard Substitution and Symmetry

Find the maximum and minimum values of the function: $$ y = \int_0^x \frac{2t-1}{t^2-t-1} \, dt \quad \text{for} \quad x \in [0,1] $$

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Problem 68

Hard Substitution and Symmetry

Approximate the function $f(x) = \int_0^x \frac{t}{1+t^4} \, dt$ with a second-degree Maclaurin polynomial and use it to calculate the approximate value of $f\left(\frac{1}{\sqrt{10}}\right)$. Estimate the error.

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Problem 69

Medium Substitution and Symmetry

Find the curvature of the curve $y = \int_0^x (1+t)\ln(1+t) \, dt$ at the point $(0,0)$.

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Problem 70

Medium Substitution and Symmetry

Find the maximum curvature of the curve $y = \int_0^x \sinh t \, dt$.

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Problem 71

Hard Substitution and Symmetry

Part 1: Approximate the function $f(x) = e^{x^2}$ with a Taylor polynomial $T_4(x)$ of degree 4 around the point $x = 0$ and estimate the error of approximation for $x \in \left[-\frac{1}{4}, \frac{1}{4}\right]$. Part 2: Calculate approximately $\int_0^{1/4} e^{x^2} \, dx$ using $\int_0^{1/4} e^{x^2} \, dx \approx \int_0^{1/4} T_4(x) \, dx$ and, using the estimate from Part 1, determine the maximum error.

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Problem 72

Hard Substitution and Symmetry

Calculate $\int_{-1}^x \frac{\arcsin t}{\sqrt{t+1}} \, dt$ and then solve the equation: $$ \int_{-1}^x \frac{\arcsin t}{\sqrt{1+t}} = 2\sqrt{x+1} \arcsin x + 4\sqrt{2} $$

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Problem 73

Hard Substitution and Symmetry

Solve the equation: $$ \int_0^x \frac{t \arctan t}{\sqrt{1+t^2}} \, dt = \sqrt{1+x^2} \arctan x $$

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Problem 74

Hard Substitution and Symmetry

Solve the equation: $$ \int_x^2 (t^4-t^2)^{-1/2} \, dt = \frac{\pi}{3} $$

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Problem 75

Medium Substitution and Symmetry

Solve the equation: $$ \int_{\ln 4}^x \frac{dt}{\sqrt{e^t-4}} = \frac{\pi}{4} $$

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Problem 76

Hard Substitution and Symmetry

If $f(x) = \arctan\frac{1}{x} - \frac{1}{1+x^2}$ and $I_n = \int_0^n f(x) \, dx$, $J_n = \int_{-n}^0 f(x) \, dx$, find: $$ \lim_{n \to \infty} \frac{I_n}{J_n} $$

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Problem 77

Hard Substitution and Symmetry

Find: $$ \lim_{t \to \infty} \frac{\int_1^t \left(\arctan\frac{x+1}{x-1} - \frac{\pi}{4}\right) dx}{\ln t} $$

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Problem 78

Medium Substitution and Symmetry

The function $f(x)$ is continuous and strictly increasing on $[0, a]$ with $f(0) = 0$. Let $g(x)$ be its inverse function and let the function $$ F(x) = \int_0^x f(t) \, dt + \int_0^{f(x)} g(t) \, dt - xf(x) $$ be defined on $[0, a]$. Prove that $F(x)$ is differentiable and that $F'(x) = 0$.

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Problem 79

Medium Substitution and Symmetry

If $f(x)$ is continuous and positive, prove that the function $$ \varphi(x) = \frac{\int_0^x t f(t) \, dt}{\int_0^x f(t) \, dt} $$ is increasing on the interval $[0, \infty)$.

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Problem 80

Hard Substitution and Symmetry

Determine the limit of the sequence whose general term is: $$ a_n = \frac{1}{n^n} \lim_{A \to \infty} \int_0^A x^n e^{-x} \, dx $$

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Problem 81

Medium Special Techniques

Calculate the integral: $\int_0^1 \frac{dx}{\sqrt{x}}$

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Problem 82

Medium Special Techniques

Calculate the integral: $$ \int_{-\infty}^{\infty} \frac{1}{1+x^2} \, dx $$

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Problem 83

Hard Special Techniques

Investigate the convergence of the integral and find its value: $$ \int_0^1 \frac{dx}{x^\alpha} $$

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Problem 84

Hard Special Techniques

Investigate the convergence of the integral and find its value: $$ \int_1^\infty \frac{dx}{x^\alpha} $$

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Problem 85

Hard Special Techniques

Investigate the convergence of the integral and find its value: $$ \int_a^b \frac{dx}{(b-x)^\alpha} \quad (a \neq b) $$

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Problem 86

Medium Special Techniques

Calculate the improper integral (or establish its divergence): $$ \int_0^5 \frac{dx}{x^7} $$

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Problem 87

Medium Special Techniques

Calculate the improper integral (or establish its divergence): $$ \int_0^e x|\ln x| \, dx $$

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Problem 88

Medium Special Techniques

Calculate the improper integral: $$ \int_0^1 \frac{dx}{\sqrt{1-x^2}} $$

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Problem 89

Hard Special Techniques

Calculate the improper integral: $$ \int_0^1 \frac{x+1}{(x-1)^{2/3}} \, dx $$

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Problem 90

Medium Special Techniques

Calculate the improper integral: $$ \int_0^2 \frac{dx}{x^2-3x+2} $$

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Problem 91

Medium Special Techniques

Calculate the improper integral: $$ \int_0^2 \frac{dx}{\sqrt[3]{(1-x)^2}} $$

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Problem 92

Medium Special Techniques

Calculate the improper integral: $$ \int_{-1}^1 \frac{dx}{\sqrt[3]{x}} $$

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Problem 93

Medium Special Techniques

Calculate the improper integral: $$ \int_0^{1/2} \frac{dx}{x\ln^2 x} $$

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Problem 94

Medium Special Techniques

Calculate the improper integral: $$ \int_a^\infty \frac{dx}{x\ln^2 x} $$ (where $a > 1$)

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Problem 95

Medium Special Techniques

Calculate the improper integral (or establish its divergence): $$ \int_2^\infty \cos x \, dx $$

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Problem 96

Medium Special Techniques

Calculate the improper integral: $$ \int_{-1}^\infty e^{-kx} \, dx $$

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Problem 97

Hard Special Techniques

Calculate the improper integral: $$ \int_0^\infty e^{-ax}\cos(bx) \, dx $$

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Problem 98

Hard Special Techniques

Calculate the improper integral: $$ \int_0^\infty e^{-ax}\sin(bx) \, dx $$

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Problem 99

Hard Special Techniques

Calculate the improper integral: $$ \int_1^\infty \frac{x\ln x}{(1+x^2)^2} \, dx $$

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Problem 100

Hard Special Techniques

Calculate the improper integral: $$ \int_0^{\pi/4} \sqrt{\tan x} \, dx $$

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Problem 101

Hard Special Techniques

Calculate the improper integral: $$ \int_1^\infty \frac{dt}{t\sqrt{t^2+a^2}} \quad (a \neq 0) $$

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Problem 102

Medium Special Techniques

Investigate the convergence of the integral: $$ \int_0^\infty \frac{\sin x \cdot \arctan x}{1+x^2} \, dx $$

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Problem 103

Medium Special Techniques

Investigate the convergence of the integral: $$ \int_1^\infty \frac{2x^2-3x+2}{\sqrt{x^8+3x^6+5}} \, dx $$

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Problem 104

Hard Special Techniques

Investigate the convergence of the integral: $$ \int_{-1}^\infty \frac{dx}{x^2+\sqrt[y]{x^4+1}} $$

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Problem 105

Medium Special Techniques

Investigate the convergence of the integral: $$ \int_1^\infty \frac{1}{xe^x} \, dx $$

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Problem 106

Medium Special Techniques

Investigate the convergence of the integral: $$ \int_0^\infty \frac{x^a \arctan x}{c+x^b} \, dx \quad (b>0, c>0) $$

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Problem 107

Hard Special Techniques

Show that the Beta function $$ B(p, q) = \int_0^1 x^{p-1}(1-x)^{q-1} \, dx $$ converges for $p > 0$ and $q > 0$.

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Problem 108

Hard Special Techniques

Show that the Gamma function $$ \Gamma(p) = \int_0^\infty x^{p-1} e^{-x} \, dx $$ converges for $p > 0$.

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Problem 109

Hard Special Techniques

Prove that $$ \Gamma(p+1) = \int_0^\infty x^p e^{-x} \, dx = p! $$ where $p$ is a natural number.

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Problem 110

Medium Special Techniques

Prove that: $$ \int_0^{\infty} x^{2n+1} e^{-x^2} \, dx = \frac{n!}{2} $$ where $n$ is a natural number.

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Problem 111

Medium Special Techniques

Prove that: $$ \int_0^{\infty} f\left(x+\frac{a^2}{x}\right) \, dx = \int_0^{\infty} f\left(\sqrt{x^2+4a^2}\right) \, dx, \quad (a>0) $$

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Problem 112

Medium Special Techniques

Calculate the integrals: $$ J_1=\int_0^{\infty} \frac{dx}{(x^2+a^2)(x^2+b^2)} \quad \text{and} \quad J_2=\int_0^{\infty} \frac{x^2 \, dx}{(x^2+a^2)(x^2+b^2)} $$ where $a \neq 0$ and $a + b \neq 0$.

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Problem 113

Medium Special Techniques

Prove that: $$ \int_0^{\frac{\pi}{2}} \frac{\sin (2k+1) \theta}{\sin \theta} \, d\theta = \frac{\pi}{2}, \quad (k=0,1,2,\ldots) $$

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Problem 114

Medium Special Techniques

Prove that: $$ \int_{-a}^a |x|^x(\ln |x|+1) \, dx = 2\sinh(a \ln a), \quad (a>0) $$

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Problem 115

Medium Special Techniques

Estimate the value of the integral: $$ \int_0^1 \sqrt{3+x^2} \, dx $$

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Problem 116

Medium Special Techniques

Estimate the value of the integral: $$ \int_0^{\infty} \frac{\cos ax}{4+x^2} \, dx $$

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Problem 117

Hard Special Techniques

Estimate the value of the integral $$ \int_0^1 \frac{dx}{\sqrt{2-x-x^2}} $$ by finding upper and lower bounds, then calculate its exact value.

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Problem 118

Medium Special Techniques

Prove that $$ \int_0^2 e^{x^2-x} \, dx \in \left(\frac{2}{\sqrt{e}}, 2e^2\right) $$

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Problem 119

Hard Special Techniques

Estimate the value of the integral $$ \int_{\frac{\sqrt{3}}{3}}^{\sqrt{3}} x \arctan x \, dx $$ then find the exact value and compare the results.

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Problem 120

Medium Special Techniques

Which is larger: $$ I_1 = \int_0^1 x^2 \sin^2 x \, dx \quad \text{or} \quad I_2 = \int_0^1 x \sin^2 x \, dx $$

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Problem 121

Medium Special Techniques

Which is larger: $$ I_a = \int_1^4 a^x \arctan x \, dx \quad \text{or} \quad I_e = \int_1^4 e^x \arctan x \, dx $$

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Problem 122

Medium Special Techniques

Which is larger: $$ I_1 = \int_1^2 3^{x^2} \, dx \quad \text{or} \quad I_2 = \int_1^2 3^{x^3} \, dx $$

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Problem 123

Medium Special Techniques

Which is larger: $$ I_n = \int_0^{\frac{\pi}{2}} \sin^n x \, dx \quad \text{or} \quad I_{n+1} = \int_0^{\frac{\pi}{2}} \sin^{n+1} x \, dx, \quad (n \in \mathbb{N}) $$

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Problem 124

Medium Special Techniques

Find the average value of the function on the given interval: $$ y = \ln x, \quad [1, e] $$

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Problem 125

Medium Special Techniques

Find the average value of the function on the given interval: $$ y = \sin^4 x, \quad [0, \pi] $$

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Problem 126

Medium Special Techniques

The average value of a function $f(x)$ on the interval $[0, \infty)$ is defined as $$ \lim_{t \to \infty} \frac{1}{t} \int_0^t f(x) \, dx $$ Find the average value of the function $f(x) = \arctan x$ on the interval $[0, \infty)$.

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Problem 127

Medium Special Techniques

Determine the sign of the integral: $$ I = \int_1^e \frac{\arctan(\ln x)}{x} \, dx $$

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Problem 128

Medium Special Techniques

Estimate the value of the integral: $$ I = \int_0^{100} \frac{1}{(x+100)e^x} \, dx $$

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Problem 129

Medium Special Techniques

Estimate the value of the integral: $$ I = \int_{100}^{200} \frac{\sin \pi x}{x} \, dx $$

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