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#