Question

I need help in evaluating the integral of the following:

[(cos(x))/((2 + sin(x))^2)]dx, from x=(-pi/2) to (pi/2).
I evaluated using u-substitution, where u = (2 + sin(x)) and got a final answer of: 0.
WolframAlpha got: 2/3

I have reworked the problem a few times, and cannot determine what I'm doing wrong. Can someone do this problem, verify which answer (if either of them) is correct, and if I am wrong, work through the problem with me?

Any and all help is greatly appreciated! :)

Answer 1

so its \[\int\limits_{-\Pi/2}^{\Pi/2}\frac{ \cos x }{ 2+\sin x^{2} }\], right?

Answer 2

Very close! It may not matter, but the bottom looks like this: (2 + sin(x))^2. (Squared is separated by parentheses)

Answer 3

oh kk sorry

Answer 4

\[\int\limits_{\frac{- \pi }{ 2 }}^{\frac{ \pi }{ 2 }}\left( 2+\sin x \right)^{-2}\cos x~dx\]
\[=\frac{ -1 }{ 2+\sin x },from ~\frac{ -\pi }{ 2 }\rightarrow \frac{ \pi }{ 2 }\]
now you can solve.
i am leaving now.

Answer 5

so its I=- 1/(2+sinx) from -PI/2, PI/2
-1/(2+1)-(-1/2-1)=-1/3+1=2/3

Answer 6

just careful with the signs

Answer 7

Oh, I see my mistake :D *eraser flying....

Answer 8

xD

Answer 9

...I'm almost done reworking:)

Answer 10

ah kk nice

Answer 11

Man do I feel stupid now.... :) I got it, +2/3. Thank you very much!