Expert: Paul Klarreich - 3/21/2009

Question
QUESTION: Hello Paul Klarreich. My question is how do you integrate this thing:

(2sinθ +1)^3

Do you first expand the whole thing and then later use reduction formula to integrate it or you just integrate it without expanding it.

ANSWER: Subject:  Integration of trig functions
Question:  Hello Paul Klarreich. My question is how do you integrate this thing:

(2sin t +1)^3    << changed to a 't'.

Do you first expand the whole thing and then later use reduction formula to integrate it or you just integrate it without expanding it.
................................
Yes,  you must expand, since you do not have a cos t available for  u-substitution.


(2sin t +1)^3 =

8 sin^3(t) + 12 sin^2(t) + 6 sin t + 1

Now you deal with each term:

First term:

sin^3(t) = sin t(sin^2(t)) = sin t(1 - cos^2(t) ), and do a u-substitution.

Second term:
         1 - cos(2t)  
sin^2(t) = ------------  << half-angle reduction.
         2

Third and fourth terms will be easy.  Suggestion (but you didn't hear it from me)

Check your answer with:

http://integrals.wolfram.com/index.jsp

(you don't always get the same form, but that just makes it more fun.)



---------- FOLLOW-UP ----------

QUESTION: I got :

-8/3{cosθ +cos^3θ }+3{θ -sin(2θ)}-6cosθ +θ  


Is this correct ?

Answer
Questioner:  nyasha
Private: no
Subject:   

 
Question:  
QUESTION: Hello Paul Klarreich. My question is how do you integrate this thing:

(2sinθ +1)^3

Do you first expand the whole thing and then later use reduction formula to integrate it or you just integrate it without expanding it.

ANSWER: Subject:  Integration of trig functions
Question:  Hello Paul Klarreich. My question is how do you integrate this thing:

(2sin t +1)^3    << changed to a 't'.

Do you first expand the whole thing and then later use reduction formula to integrate it or you just integrate it without expanding it.
................................
Yes,  you must expand, since you do not have a cos t available for  u-substitution.


(2sin t +1)^3 =

8 sin^3(t) + 12 sin^2(t) + 6 sin t + 1

Now you deal with each term:

First term:

sin^3(t) = sin t(sin^2(t)) = sin t(1 - cos^2(t) ), and do a u-substitution.

Second term:
         1 - cos(2t)  
sin^2(t) = ------------  << half-angle reduction.
         2

Third and fourth terms will be easy.  Suggestion (but you didn't hear it from me)

Check your answer with:

http://integrals.wolfram.com/index.jsp

(you don't always get the same form, but that just makes it more fun.)



---------- FOLLOW-UP ----------

QUESTION: I got :

-8/3{cosθ +cos^3θ }+3{θ -sin(2θ)}-6cosθ +θ  


Is this correct ?
--------------------------------
Probably, but I don't have time to check it out.