about the integral of (cotx)^3 why cant i add the power and divide by the new power and the derivative of cotx here

Question

as far as i know that any first degree function with a power the integral of it is adding power and division by the new power is that right?

Answer 1

If you differentiate (cot⁴ x)/4 you will get (cot³ x)(-csc x). The extra -csc x appears because of the chain rule. Note, however, it is missing in the integrant.

 

The correct way is to rewrite using the quotient identity and then the Pythagorean identity. Then we wind up being able to do a u-substitution.

 

∫ cot³ x dx

 

=∫ (cos³ x)/(sin³ x) dx

 

=∫ [(1-sin² x)cos x]/(sin³ x) dx

 

=  ∫ (1-u²)/u³ du : Using u = sin x , du = cos x dx

 

= ∫ (u^-3 - 1/u) du

 

= -u^-2/2 - ln|u| + C

 

 

= -(csc² x)/2 - ln|sin x| + C