Question

find integral[ x/(1+sin(x))

Answer 1

I would try the Weierstrass substitution

Answer 2

what about the "x" in the numerator ?

Answer 3

\[x=\frac{\tan^{-1}(t)}{2}\]

it can be done but it is really nasty

you could also try multiplying top and bottom by \[1-\sin(x)\]

and write in terms of sec(x) and tan(x)

Answer 4

ya i tried but the problem always is the "x" in the numerator how to get rid of it ?

Answer 5

well changing an algebric part to an inverse function not advisory in integration !

Answer 6

if you use my second suggestion you can integrate by parts to get rid of the x

Answer 7

following that i get an expression

Answer 8

\[\[\int\limits\limits_{}^{}\sec ^{-1}t dt\]\]

Answer 9

\[\int\frac{x}{1+\sin(x)}dx\]

\[\int\frac{x}{1+\sin(x)}\frac{1-\sin(x)}{1-\sin(x)}dx\]

\[\int\frac{x(1-\sin(x))}{1-\sin^2(x)}dx\]

\[\int\frac{x(1-\sin(x))}{\cos^2(x)}dx\]

\[\int\frac{x-x\sin(x)}{\cos^2(x)}dx\]


\[\int x\cdot\sec^2(x)-x\cdot\sec(x)\tan(x)dx\]