Integral of the power products, solve the following integral: (sin^3sqrt(x))/sqrt(x)dx ?

1 Answer
Apr 5, 2018

I=-2cos(sqrt(x))+2/3cos^3(sqrt(x))+C

Explanation:

We want to solve

I=intsin^3(sqrt(x))/sqrt(x)dx

Make a substitution u=sqrt(x)=>du=1/(2sqrt(x))dx

I=intsin^3(u)/sqrt(x)*2sqrt(x)du=2intsin^3(u)du

By the Pythagorean trig identity

I=2intsin(u)(1-cos^2(u))du

color(white)(I)=2intsin(u)du-2intsin(u)cos^2(u)du

For the second integral substitute s=cos(u)=>ds=-sin(u)du

I=2intsin(u)du+2ints^2ds

color(white)(I)=-2cos(u)+2/3s^3+C

Substitute back s=cos(u) and u=sqrt(x)

I=-2cos(sqrt(x))+2/3cos^3(sqrt(x))+C