What is the integral from-pi to pi of (x^2)/(1+x^2) ?

The integral from-pi to pi of (x^2)/(1+x^2) is -2arctan(pi)+2pi

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integral from-pi to pi of (x^2)/(1+x^2)

\int_{- π}^{π} \frac{x ^{2}}{1 + x ^{2}} d x

- 2 arctan (π) + 2 π

Decimal

3.75793 \dots

Solution steps

\int_{- π}^{π} \frac{x ^{2}}{1 + x ^{2}} d x

\frac{x ^{2}}{1 + x ^{2}} = - \frac{1}{1 + x ^{2}} + 1

= \int_{- π}^{π} - \frac{1}{1 + x ^{2}} + 1 d x

Apply the Sum Rule:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

= - \int_{- π}^{π} \frac{1}{1 + x ^{2}} d x + \int_{- π}^{π} 1 d x

\int_{- π}^{π} \frac{1}{1 + x ^{2}} d x = 2 arctan (π)

\int_{- π}^{π} 1 d x = 2 π

= - 2 arctan (π) + 2 π

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What is the integral from-pi to pi of (x^2)/(1+x^2) ?
The integral from-pi to pi of (x^2)/(1+x^2) is -2arctan(pi)+2pi