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

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

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

integral of 3x^2arctan(x)

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

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

Solution steps

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

Apply Integration By Parts

= 3 (\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)

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

Simplify

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

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

Add a constant to the solution

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

\int 8 cos (π x) d x

\int \frac{1}{x 144 x ^{2} - 1} d x

\int \frac{4 x}{x ^{2} + 3} d x

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

\int cos (x) + \frac{1}{6} x d x

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