Evaluate the integral: integral (2x - 3 sin x + cos x) dx
Evaluate the integral: $$ \int (2x - 3\sin x + \cos x) \, dx. $$
Explanation of Method
Integration is linear: the integral of a sum (or constant multiple) of functions is the sum of the individual integrals. Here the integrand mixes a polynomial term with two standard trig functions, so each piece is integrated separately using its own known antiderivative.
- Split the integral term by term using linearity.
- Apply the power rule to the polynomial term.
- Apply the standard antiderivatives of $\sin x$ and $\cos x$.
- Combine the results and add the constant of integration.
Step 1 · Evaluate each term
The integral can be separated into three simpler integrals:
$$ \int (2x - 3\sin x + \cos x)\,dx = \int 2x\,dx - \int 3\sin x\,dx + \int \cos x\,dx. $$
First term: $\displaystyle \int 2x\,dx$. Using the power rule for integration, $\int x^n\,dx = \dfrac{x^{n+1}}{n+1}$ (for $n\neq -1$), we have:
$$ \int 2x\,dx = 2\cdot \frac{x^2}{2} = x^2. $$
Second term: $\displaystyle \int -3\sin x\,dx$. Using the fact that the integral of $\sin x$ is $-\cos x$:
$$ \int -3\sin x\,dx = -3\int \sin x\,dx = -3(-\cos x) = 3\cos x. $$
Third term: $\displaystyle \int \cos x\,dx$. Using the fact that the integral of $\cos x$ is $\sin x$:
$$ \int \cos x\,dx = \sin x. $$
Step 2 · Combine the results
Combine all three terms:
$$ \int (2x - 3\sin x + \cos x)\,dx = x^2 + 3\cos x + \sin x + C, $$
where $C$ is the constant of integration.
Summary Table
| Term | Antiderivative |
|---|---|
| $2x$ | $x^2$ |
| $-3\sin x$ | $3\cos x$ |
| $\cos x$ | $\sin x$ |
Additional Notes
- The integrand is defined for all real $x$, since polynomials, $\sin x$, and $\cos x$ are defined everywhere.
- Because integration is linear, each term's constant coefficient carries straight through to its antiderivative — no product or chain rule considerations are needed here.
- The result can be checked by differentiating $x^2 + 3\cos x + \sin x$ and confirming it returns the original integrand.
Result
The solution to the integral is: