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

The integral of (x^2)/(2(x^2+1)) is 1/2 (-arctan(x)+x)+C

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integral of (x^2)/(2(x^2+1))

\int \frac{x ^{2}}{2 ( x ^{2} + 1 )} d x

\frac{1}{2} (- arctan (x) + x) + C

Solution steps

\int \frac{x ^{2}}{2 ( x ^{2} + 1 )} d x

Take the constant out:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= \frac{1}{2} \cdot \int \frac{x ^{2}}{x ^{2} + 1} d x

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

= \frac{1}{2} \cdot \int - \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

= \frac{1}{2} (- \int \frac{1}{x ^{2} + 1} d x + \int 1 d x)

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

\int 1 d x = x

= \frac{1}{2} (- arctan (x) + x)

Add a constant to the solution

= \frac{1}{2} (- arctan (x) + x) + C

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