Question

When evaluating the integral of "(cosx)/((2 + sinx)^2)(dx)", with upper and lower lower limits of Pi/2 & -Pi/2 respectively, what is the answer and what are the steps? I know how to work these problems, generally, but am having a some hiccups on a couple questions.
Any and all help is greatly appreciated! :)

Answer 1

So far I have found: u = (2 + sinx)^2
du = 2(2 + sinx)(cosx)(dx)

Answer 2

\[\int\limits_{\frac{ -\pi }{ 2 }}^{\frac{ \pi }{ 2 }}\frac{ \cos x }{\left( 2+\sin x \right)^2 }dx\]

Answer 3

Also, cos(dx) = du/[2(2 + sinx)]

Answer 4

Correct!

Answer 5

it is better if you substitute (2+sinx)=t and than find the derivative.

Answer 6

put 2+sin x=t
cos x dx=dt
\[when~x=-\pi/2,t=2+\sin(-\pi/2) ,t=2-\sin \frac{ \pi }{2 }=2-1=1\]
\[when~ x=\frac{ \pi }{2 },t=2+\sin \frac{ \pi }{2 }=2+1=3\]
solve.

Answer 7

\[I=\int\limits_{1}^{3} t ^{-2}~dt\]=?

Answer 8

...when do you use the anti derivative of the integral, after u (or t) substitution, and then subtract the anti.deriv. with the lower limit plugged into it, from the anti.deriv. With the upper limit plugged into it?