Question

integration of (cos(x))/(sin(x)+(sin(x))^2)

Answer 1

\[\frac{\cos(x)}{\sin(x) +(\sin(x))^2}\]

Answer 2

let t=sin x

Answer 3

I think:

Put:

\[1 + sinx = t\]

Answer 4

@KonradZuse
\[dt = \cos x \ dx\]

Answer 5

\[\frac{cosx}{sinx(1 +sinx)}dx\]

Answer 6

Put:
\[1 + sinx = u\]

\[sinx = (u-1)\] -------------1

\[du = cosxdx\]

\[\int\limits_{}^{}\frac{du}{u(u-1)}\]

Answer 7

interesting.

Answer 8

so then do we do u^2 - u?

Answer 9

then it owuld be u^3/3 - ln|u| +c?

Answer 10

Now use completion of square method..

Answer 11

just notice
\[\frac{1}{u(u-1)}=\frac{1}{u-1}-\frac{1}{u}\]

Answer 12

Yeah mukushla is right you can use this it will become easy to solve..

Answer 13

Idk how we would do completing the square. So my method was wrong huh...

Answer 14

No go by mukushula method..
He has just simplified it for you above..

Answer 15

I guess it would be ln|u-1| - ln|u| + c?

Answer 16

right

Answer 17

I see what I did and why it's not right, but you couldn't just put them back together then huh...

Answer 18

ln|sin(x)| - ln|1+sin(x)| +c

Answer 19

is that correct?

Answer 20

yes

Answer 21

Thanks :)

Answer 22

Wolfram gave me a super weird answer....

Answer 23

Do not forget to replace u by (1 + sinx) ..

Answer 24

http://www.wolframalpha.com/input/?i=integration+of+%28cos%28x%29%29%2F%28sin%28x%29%2B%28sin%28x%29%29%5E2%29

Yes I did water :)