Improving Integration Skills: Solving a Tricky Integral Problem

[student93]

Homework Statement

Question is attached in this post.

Homework Equations

Question is attached in this post.

The Attempt at a Solution

I set the integral as ∫(cos^1/2(x))(1-cos^2(x))(sinx) dx from 0 to π/2

Next I used u-substitution where I let u=cosx and du=-sinxdx

I solved out the integral and ended up getting 7/6 as my answer, however the correct answer is 8/21. Was my approach incorrect?

 

[PhysicoRaj]

You might be wrong in applying limits and simplifying. Check it. :)

 

[SteamKing]

Obviously, something went wrong, but we can't say unless you show the details of your work.

 

[student93]

You might be wrong in applying limits and simplifying. Check it. :)

Could you please tell me how I used the wrong limits, and what the actual limits are? (This question is really frustrating me and I have an exam in a few hours etc.).

 

[PhysicoRaj]

Have you changed the limits according to your substitution?
The limits given are for x. Calculate them for u.
OR
have you replaced u by an x function?

 

[student93]

Obviously, something went wrong, but we can't say unless you show the details of your work.

∫cosx^1/2(1-cos^2x)sinxdx

u=cosx
du=-sinxdx

-1∫u^1/2 - u

= -1((u^3/2)(2/3))(u^2/2)

= -1((cos^3/2(x))(2/3))cos^2(x)/2 -> Upper Limit = π/2 and Lower Limit =0

My answer comes out to 7/6, is the method that I used even correct? (Or should I have had used another method?)

 

[PhysicoRaj]

∫cosx^1/2(1-cos^2x)sinxdx

u=cosx
du=-sinxdx

-1∫u^1/2 - u

= -1((u^3/2)(2/3))(u^2/2)

= -1((cos^3/2(x))(2/3))cos^2(x)/2 -> Upper Limit = π/2 and Lower Limit =0

My answer comes out to 7/6, is the method that I used even correct? (Or should I have had used another method?)

-1∫u^1/2 - u ...you have this wrong.

 

[student93]

-1∫u^1/2 - u ...you have this wrong.

How? I'm just really confused lol, shouldn't the be correct? Did I make a simple mistake?

 

[PhysicoRaj]

√u(1-u²)
=√u-u^2+0.5
=√u-u^5/2

 

[student93]

√u(1-u²)
=√u-u^2+0.5
=√u-u^5/2

Thanks, I'll try to see if I can get the correct answer now.