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

The integral of (2x^3+x)arctan(x) is 1/2 x^4arctan(x)-(x^3)/6+1/2 x^2arctan(x)+C

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

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

\frac{1}{2} x^{4} arctan (x) - \frac{x ^{3}}{6} + \frac{1}{2} x^{2} arctan (x) + C

Solution steps

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

Expand

(2 x^{3} + x) arctan (x) : 2 x^{3} arctan (x) + x arctan (x)

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

\int 2 x^{3} arctan (x) d x = 2 (\frac{1}{4} x^{4} arctan (x) - \frac{1}{4} (\frac{x ^{3}}{3} - x + arctan (x)))

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

= 2 (\frac{1}{4} x^{4} arctan (x) - \frac{1}{4} (\frac{x ^{3}}{3} - x + arctan (x))) + \frac{1}{2} x^{2} arctan (x) - \frac{1}{2} (- arctan (x) + x)

Simplify

2 (\frac{1}{4} x^{4} arctan (x) - \frac{1}{4} (\frac{x ^{3}}{3} - x + arctan (x))) + \frac{1}{2} x^{2} arctan (x) - \frac{1}{2} (- arctan (x) + x) : \frac{1}{2} x^{4} arctan (x) - \frac{x ^{3}}{6} + \frac{1}{2} x^{2} arctan (x)

= \frac{1}{2} x^{4} arctan (x) - \frac{x ^{3}}{6} + \frac{1}{2} x^{2} arctan (x)

Add a constant to the solution

= \frac{1}{2} x^{4} arctan (x) - \frac{x ^{3}}{6} + \frac{1}{2} x^{2} arctan (x) + C

What is the integral of (2x^3+x)arctan(x) ?
The integral of (2x^3+x)arctan(x) is 1/2 x^4arctan(x)-(x^3)/6+1/2 x^2arctan(x)+C