integrating

Question

when integrating ∫sin³xdx, a good first move would be to:

 

a) use u-substitution, letting u=sinx, and then integrating ∫u³du

 

b) use u-substitution, letting u=sin²x

 

c) rewrite the problem as ∫sinx(1-cos² x)dx

 

Answer 1

∫sin³x dx = ∫(sin²x)(sinx) dx

             

              = ∫(1 - cos²x)(sinx) dx

 

 

Using u-substitution

 

Let   u = cosx

       du = -sinx dx

      -du = sinx dx

 

 

Plugging in these values, you get

 

∫(1 - u²)(-du) =

 

- ∫(1 - u²) du

 

 

This integral is now easy to work with.  So the best way to start is    choice (c).