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

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

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

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

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

Solution steps

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

Take the constant out:

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

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

Apply Integration By Parts

= 3 (\frac{1}{3} x^{3} arctan (x^{3}) - \int \frac{x ^{5}}{x ^{6} + 1} d x)

\int \frac{x ^{5}}{x ^{6} + 1} d x = \frac{1}{6} ln x^{6} + 1

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

Simplify

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

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

Add a constant to the solution

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

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