Integral of the power products, solve the following integral: #(sin^3sqrt(x))/sqrt(x)dx# ?
1 Answer
Apr 5, 2018
Explanation:
We want to solve
#I=intsin^3(sqrt(x))/sqrt(x)dx#
Make a substitution
#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
#I=2intsin(u)du+2ints^2ds#
#color(white)(I)=-2cos(u)+2/3s^3+C#
Substitute back
#I=-2cos(sqrt(x))+2/3cos^3(sqrt(x))+C#