# How do you find the antiderivative of #int xsqrt(100-x^2)dx#?

##### 1 Answer

Nov 6, 2016

#### Explanation:

#I=intxsqrt(100-x^2)dx#

The best substitution to make here is

Since we have just

#I=-1/2int(-2x)sqrt(100-x^2)dx#

#I=-1/2intunderbrace(sqrt(100-x^2))_sqrtuoverbrace((-2xdx))^(du)#

#I=-1/2intsqrtudu#

This can be integrated if we write the square root using a fractional power.

#I=-1/2intu^(1/2)du#

This can be integrated using the power rule for integration, or

#I=-1/2(u^(1/2+1)/(1/2+1))+C=-1/2(u^(3/2)/(3/2))+C#

#I=-1/2(2/3)u^(3/2)+C=-u^(3/2)/3+C#

Since

#I=-(100-x^2)^(3/2)/3+C#