Finding ∫x sin^3x dx

  • Thread starter Bimpo
  • Start date
  • #1
9
0

Homework Statement



Finding ∫x sin^3x dx

Homework Equations



I don't think this is needed

The Attempt at a Solution


∫x sin^3x dx using integration by parts u = x, du = dx, dv = sin^3x, v = 1/3cos^3x - cosx
= (x)(1/3cos^3x - cosx) - ∫1/3cos^3x - ∫cosx dx
= (x)(1/3cos^3x - cosx) - 1/3∫cos^3x dx + (sinx)
= (x)(1/3cos^3x - cosx) - 1/3∫cos^2xcosx dx + (sinx)
= (x)(1/3cos^3x - cosx) - 1/3∫(1-sin^2x)(cosx) dx + (sinx)
= (x)(1/3cos^3x - cosx) - (1/3∫cosx dx - ∫cosxsin^2x dx) + (sinx)
= (x)(1/3cos^3x - cosx) - (1/3 (-sinx) - ∫cosxsin^2x dx) + (sinx)
using subsitution u = sinx, du = cosx dx
= (x)(1/3cos^3x - cosx) - (1/3 (-sinx) - ∫u^2 du) + (sinx)
= (x)(1/3cos^3x - cosx) - (1/3 (-sinx) - 1/3u^3) + (sinx)
then subsitute back
= (x)(1/3cos^3x - cosx) - (1/3 (-sinx) - 1/3sin^3x) + (sinx) + C

Anything I did wrong?
Anyway to do this much easier/faster?
 

Answers and Replies

  • #2
Don't integrate by parts. Not yet, at least.

Use the half-angle identity:
sin2x = 1/2(1 - cos(2x))
 
  • #3
9
0
Ok, I do that then
I'll get something like:

= 1/2∫(1-cos2x)xsinx dx
= 1/2∫xsinx - ∫xsinxcos2x dx <-- I get stuck here
 
  • #4
vela
Staff Emeritus
Science Advisor
Homework Helper
Education Advisor
14,793
1,379
You can always check your answer by differentiating it and seeing if you recover the integrand.
 
  • #5
SammyS
Staff Emeritus
Science Advisor
Homework Helper
Gold Member
11,325
1,008

Homework Statement



Finding ∫x sin^3x dx

The Attempt at a Solution


∫x sin^3x dx using integration by parts u = x, du = dx, dv = sin^3x  dx, v = 1/3cos^3x - cosx
To find v by integrating, I would rewrite sin3 x as

sin3 x = (1 - cos2 x)sin  x
= sin  x - sin  x   cos2 x​

This is fairly easy to integrate.
 

Related Threads on Finding ∫x sin^3x dx

  • Last Post
Replies
13
Views
3K
  • Last Post
Replies
10
Views
4K
  • Last Post
Replies
2
Views
15K
  • Last Post
Replies
5
Views
9K
  • Last Post
Replies
3
Views
2K
  • Last Post
Replies
1
Views
1K
  • Last Post
Replies
3
Views
18K
Replies
1
Views
6K
Replies
1
Views
2K
  • Last Post
Replies
15
Views
3K
Top