What is the integral of (2x^2+3)arctan(x) ?

The integral of (2x^2+3)arctan(x) is arctan(x)((2x^3)/3+3x)-1/6 (2x^2+2+7ln|x^2+1|)+C

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integral of (2x^2+3)arctan(x)

\int (2 x^{2} + 3) arctan (x) d x

arctan (x) (\frac{2 x ^{3}}{3} + 3 x) - \frac{1}{6} (2 x^{2} + 2 + 7 ln x^{2} + 1) + C

Solution steps

\int (2 x^{2} + 3) arctan (x) d x

Apply Integration By Parts

: arctan (x) (\frac{2 x ^{3}}{3} + 3 x) - \int \frac{2 x ^{3} + 9 x}{3 ( x ^{2} + 1 )} d x

= arctan (x) (\frac{2 x ^{3}}{3} + 3 x) - \int \frac{2 x ^{3} + 9 x}{3 ( x ^{2} + 1 )} d x

\int \frac{2 x ^{3} + 9 x}{3 ( x ^{2} + 1 )} d x = \frac{1}{6} (2 x^{2} + 2 + 7 ln x^{2} + 1)

= arctan (x) (\frac{2 x ^{3}}{3} + 3 x) - \frac{1}{6} (2 x^{2} + 2 + 7 ln x^{2} + 1)

Add a constant to the solution

= arctan (x) (\frac{2 x ^{3}}{3} + 3 x) - \frac{1}{6} (2 x^{2} + 2 + 7 ln x^{2} + 1) + C

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What is the integral of (2x^2+3)arctan(x) ?
The integral of (2x^2+3)arctan(x) is arctan(x)((2x^3)/3+3x)-1/6 (2x^2+2+7ln|x^2+1|)+C