Evaluate the integral: I = integral (x²)/(1 + x²) dx
Evaluate the integral: $$ I = \int \frac{x^2}{1 + x^2} \, dx. $$
Explanation of Method
Adding and subtracting a constant in the numerator so that it matches the denominator is a standard trick for improper rational integrands with an irreducible quadratic denominator. Writing $x^2 = (1+x^2)-1$ splits the fraction into a constant term and the standard arctangent integral $\int \frac{dx}{1+x^2}$.
- Rewrite the numerator in terms of the denominator plus a remainder.
- Split the fraction into a constant term and a standard form.
- Integrate each piece separately.
- Combine the results and add the constant of integration.
Step 1 · Simplify the integrand
Rewrite the numerator $x^2$ as:
$$ x^2 = (1+x^2) - 1. $$
Substituting this into the integrand:
$$ \frac{x^2}{1+x^2} = \frac{(1+x^2)-1}{1+x^2} = 1 - \frac{1}{1+x^2}. $$
Thus, the integral becomes:
$$ I = \int \left(1 - \frac{1}{1+x^2}\right) dx. $$
Step 2 · Split the integral
Split the integral into two simpler integrals:
$$ I = \int 1\,dx - \int \frac{1}{1+x^2}\,dx. $$
Step 3 · Evaluate each integral
The first integral is straightforward:
$$ \int 1\,dx = x. $$
The second integral, $\displaystyle \int \frac{1}{1+x^2}\,dx$, is a standard integral:
$$ \int \frac{1}{1+x^2}\,dx = \arctan(x). $$
Step 4 · Combine the results
Combine the results of the two integrals:
$$ I = x - \arctan(x) + C, $$
where $C$ is the constant of integration.
Summary Table
| Term | Type | Antiderivative |
|---|---|---|
| $1$ | Constant term | $x$ |
| $1/(1+x^2)$ | Standard arctangent form | $-\arctan(x)$ (subtracted) |
Additional Notes
- The integrand is defined for all real $x$, since $1+x^2 \neq 0$ always.
- This "add and subtract" trick generalizes: any improper fraction $\frac{p(x)}{q(x)}$ with $\deg p = \deg q$ can be reduced the same way before applying a standard integral form.
- The result can be checked by differentiating $x - \arctan(x)$ and confirming it returns the original integrand.
Result
The solution to the integral is: