What is the integral from 0 to 2 of (x^2)/(x^2+1) ?

The integral from 0 to 2 of (x^2)/(x^2+1) is -arctan(2)+2

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integral from 0 to 2 of (x^2)/(x^2+1)

\int_{0}^{2} \frac{x ^{2}}{x ^{2} + 1} d x

- arctan (2) + 2

Decimal

0.89285 \dots

Solution steps

\int_{0}^{2} \frac{x ^{2}}{x ^{2} + 1} d x

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

= \int_{0}^{2} - \frac{1}{x ^{2} + 1} + 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_{0}^{2} \frac{1}{x ^{2} + 1} d x + \int_{0}^{2} 1 d x

\int_{0}^{2} \frac{1}{x ^{2} + 1} d x = arctan (2)

\int_{0}^{2} 1 d x = 2

= - arctan (2) + 2

\int_{1}^{2} 1 + \frac{9 x}{4} d x

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\int_{0}^{4} x^{2} - 7 x + 12 d x

What is the integral from 0 to 2 of (x^2)/(x^2+1) ?
The integral from 0 to 2 of (x^2)/(x^2+1) is -arctan(2)+2