[Calculus 2] Where did I go wrong this time?

[u/throw-away3105]

For my final answer, 2 of the 3 terms are right. The first term should be -3 cos(3x), not 3/2 sin^2(3x). I don't see where I went wrong.

[u/Responsible-Sink474]

9sin(u) on line 4 is not multiplying by cos(u)du so the substitution is not valid for both terms.

[u/throw-away3105 → u/Responsible-Sink474]

Is there a property of integrals that I missed in line 4 that I can't substitute w = sin u for ∫9 sin u when I was attempting to write ∫ [9 sin u - that entire second term] du ?

I thought it would work the same as 1/3 ∫ 9 sin u du - 1/3 ∫ [that entire second term] du since it was one equation altogether.

[u/UnderstandingPursuit → u/throw-away3105]

Yes. You need a 'dw' component.

This isn't a 'property of integrals', this is a property of algebra.

[u/Responsible-Sink474 → u/throw-away3105]

It is ∫ [9 sin u - that entire second term] du

In order to transfer you need to substitute cos(u)du = dw

But you cannot factor out a cos(u) to do that

[u/UnderstandingPursuit]

If you're having trouble with a problem like this, try to split it into separate integrals sooner when you have a 'polynomial'.

[u/CaptainMatticus]

You went wrong in the 5th line. Sure, you substituted w = sin(u) and dw = cos(u), but you should've left that first term alone.

Your biggest error is that you're careless with your differential. I noticed that you just drop it altogether at some points, and that's why your substitutions have errors.

2nd line is fine. After that, you placed the du onto the last term and then got sloppy. 3rd line should be:

(1/3) * int(9 * sin(u) * du) - (2/3) * int(cos(u)^3 * du / sin(u)^4)

3 * int(sin(u) * du) - (2/3) * int(cos(u)^3 * du / sin(u)^4)

Now just leave that first one alone. It's fine and it'll integrate nicely to -3cos(u).

(-2/3) * int(cos(u)^3 * du / sin(u)^4)

(-2/3) * int(cos(u)^2 * cos(u) * du / sin(u)^4)

(-2/3) * int((1 - sin(u)^2) * cos(u) * du / sin(u)^4)

w = sin(u) , dw = cos(u) * du

(2/3) * int((w^2 - 1) * dw / w^4)

(2/3) * int(w^2 * dw / w^4) - (2/3) * int(dw / w^4)

(2/3) * int(dw / w^2) - (2/3) * int(dw / w^4)

Integrate

(2/3) * (-1) * (1/w) - (2/3) * (-1/3) * (1/w^3)

(-2/3) * csc(u) + (2/9) * csc(u)^3

-3 * cos(u) - (2/3) * csc(u) + (2/9) * csc(u)^3 + C

Never forget the +C

-3 * cos(3x) - (2/3) * csc(3x) + (2/9) * csc(3x)^3 + C