Indefinite Trignometric Integral

[in the rye]

Homework Statement

∫sinxcos(x/2)dx

This isn't an actual homework problem, but one I found that I'm working on for test prep.

Homework Equations

The Attempt at a Solution

[/B]
∫sinxcos(x/2) dx = ∫sinx√((1+cosx)/2) dx

u = ½ + ½ cosx
-2 du = sinx dx

-2∫√(u) du = -2(2/3⋅u^3/2) + c

-2(2/3⋅u^3/2) + c = -4/3⋅[(1+cosx)/2)^3/2] + c
-4/3⋅[(1+cosx)/2)^3/2] = -4/3⋅(cos(x/2))^3/2 + c

I tried to check my solution by graphing my answer with theirs, but the table is displaying different values.

 

[Samy_A]

Homework Statement

∫sinxcos(x/2)dx

This isn't an actual homework problem, but one I found that I'm working on for test prep.

Homework Equations

The Attempt at a Solution

[/B]
∫sinxcos(x/2) dx = ∫sinx√((1+cosx)/2) dx

u = ½ + ½ cosx
-2 du = sinx dx

-2∫√(u) du = -2(2/3⋅u^3/2) + c

-2(2/3⋅u^3/2) + c = -4/3⋅[(1+cosx)/2)^3/2] + c
-4/3⋅[(1+cosx)/2)^3/2] = -4/3⋅(cos(x/2))^3/2 + c

I tried to check my solution by graphing my answer with theirs, but the table is displaying different values.

You have an error in the last line:
The formula you used for the substitution is $\sqrt{\frac{1+\cos x}{2}}=\cos \frac{x}{2}$

 

[in the rye]

You have an error in the last line:
The formula you used for the substitution is $\sqrt{\frac{1+\cos x}{2}}=\cos \frac{x}{2}$

Gotcha, so it's probably easier to leave it where I had it in the second line. For some reason the solution isn't working, still. Even when I just plug the integral on my graphic calculator. :/

Is there a trick to getting my square roots to look fancy like yours so I can clean it up when I post?

 

[MidgetDwarf]

there's another way, I think it is easier. Remember the product to sum trigonometric identities? Use it!

 

[Samy_A]

Gotcha, so it's probably easier to leave it where I had it in the second line. For some reason the solution isn't working, still. Even when I just plug the integral on my graphic calculator. :/

Weird, I get the same result as you, except for the erroneous square root in the final result.

@MidgetDwarf 's suggestion is also good, but as said, you really are one symbol away from the solution.

Is there a trick to getting my square roots to look fancy like yours so I can clean it up when I post?

Yes, I use $\LaTeX$. It may look difficult if you never used it, but you'll learn it quickly. It makes posts with equations much more readable, and I highly recommend the use of it.

There is a guide here, there is also a link to that guide in the bottom left of the edit box.
If you see a complex $\LaTeX$ expression and want to see how it was composed, right-click the expression written in it and select "Show Math As -> TeX Commands". A small window will appear containing the code used for that expression. Or just quote a post: the $\LaTeX$ code will in general appear between ## or $$ tags.