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

The integral of arctan(x)(x^2)/2) is d/2 (1/3 x^3arctan(x)-1/6 (x^2+1-ln|x^2+1|))+C

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integral of arctan(x)(x^2)/2)

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

\frac{d}{2} (\frac{1}{3} x^{3} arctan (x) - \frac{1}{6} (x^{2} + 1 - ln x^{2} + 1)) + C

Solution steps

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

Take the constant out:

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

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

Simplify

d \frac{1}{2} : \frac{d}{2}

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

Apply Integration By Parts

: \frac{1}{3} x^{3} arctan (x) - \int \frac{x ^{3}}{3 ( x ^{2} + 1 )} d x

= \frac{d}{2} (\frac{1}{3} x^{3} arctan (x) - \int \frac{x ^{3}}{3 ( x ^{2} + 1 )} d x)

\int \frac{x ^{3}}{3 ( x ^{2} + 1 )} d x = \frac{1}{6} (x^{2} + 1 - ln x^{2} + 1)

= \frac{d}{2} (\frac{1}{3} x^{3} arctan (x) - \frac{1}{6} (x^{2} + 1 - ln x^{2} + 1))

Add a constant to the solution

= \frac{d}{2} (\frac{1}{3} x^{3} arctan (x) - \frac{1}{6} (x^{2} + 1 - ln x^{2} + 1)) + C

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What is the integral of arctan(x)(x^2)/2) ?
The integral of arctan(x)(x^2)/2) is d/2 (1/3 x^3arctan(x)-1/6 (x^2+1-ln|x^2+1|))+C