What is the integral of x^2 1/(sqrt(9-x^2)) ?

The integral of x^2 1/(sqrt(9-x^2)) is 9/2 arcsin(1/3 x)-9/4 sin(2arcsin(1/3 x))+C

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

integral of x^2 1/(sqrt(9-x^2))

\int x^{2} \frac{1}{9 - x ^{2}} d x

\frac{9}{2} arcsin (\frac{1}{3} x) - \frac{9}{4} sin (2 arcsin (\frac{1}{3} x)) + C

Solution steps

\int x^{2} \frac{1}{9 - x ^{2}} d x

Apply Integration By Parts

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

\int 2 x arcsin (\frac{1}{3} x) d x = x^{2} arcsin (\frac{1}{3} x) - \frac{9}{2} arcsin (\frac{1}{3} x) + \frac{9}{4} sin (2 arcsin (\frac{1}{3} x))

= x^{2} arcsin (\frac{1}{3} x) - (x^{2} arcsin (\frac{1}{3} x) - \frac{9}{2} arcsin (\frac{1}{3} x) + \frac{9}{4} sin (2 arcsin (\frac{1}{3} x)))

Simplify

= \frac{9}{2} arcsin (\frac{1}{3} x) - \frac{9}{4} sin (2 arcsin (\frac{1}{3} x))

Add a constant to the solution

= \frac{9}{2} arcsin (\frac{1}{3} x) - \frac{9}{4} sin (2 arcsin (\frac{1}{3} x)) + C

What is the integral of x^2 1/(sqrt(9-x^2)) ?
The integral of x^2 1/(sqrt(9-x^2)) is 9/2 arcsin(1/3 x)-9/4 sin(2arcsin(1/3 x))+C