Question

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

Answer 1

`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`