Let the integral be: \int \sin^4(x)\cos^3(x)dx NS

Kathleen Rausch

Kathleen Rausch

Answered question

2021-12-27

Let the integral be:
sin4(x)cos3(x)dx
I have to integrate this function by changing the variable. I'm trying: u=sin(x) and so du=cos(x)dx. By rewriting the integral I get:
u4cos2(x)du
But I'm stuck here because I'm not sure there should be any expression with x left in the integral.Also I know the final answer is :
sin4xcos3xdx=sin7x7+sin5x5

Answer & Explanation

Samantha Brown

Samantha Brown

Beginner2021-12-28Added 35 answers

Hint:
Use the identity
cos2(x)=1sin2(x)
and don't forget the constant of integration!
Toni Scott

Toni Scott

Beginner2021-12-29Added 32 answers

As the exponent of cos is odd, you can set
u=sinx,du=cosxdx
If the function to integrate had been, say sin3xcos4xdx, we would have set
u=cosx,du=sinxdx
Vasquez

Vasquez

Expert2022-01-09Added 669 answers

sin4xcos3xdx=15cos2x(5cos(x)sin4x)dx
=15cos2x(sin5x)dx
Now perform an integration by parts and use the same technique for the new integral.

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