A complex one. Integrate
(3 + 2cosx) / (2 + 3cosx)^2

Question

A complex one. Integrate
(3 + 2cosx) / (2 + 3cosx)^2

shubhamsrg · Answer

write 3 as 3cos^2x + 3sin^2x and see if you can make it out further

loser66 · Answer

then, partial fraction to solve the integral,

shubhamsrg · Answer

anyone getting my method? :P

anonymous · Answer

I don't think there is any ray of hope in it :P

anonymous · Answer

i need some time.

shubhamsrg · Answer

hmm, after expressing it as 3cos^2x + 3sin^2x , take cosx common.
what i mean to say is :
numerator = cosx(3cosx + 2) + 3sin^2x

now you see something building up here? :|

anonymous · Answer

yeah yeah looks like its progressing

shubhamsrg · Answer

go on! :P
try further

anonymous · Answer

no its not working

shubhamsrg · Answer

what if I represent numerator like this :

(2+ 3cosx)* d(sinx)  - sinx * d(2 + 3cosx)

anonymous · Answer

how ??

shubhamsrg · Answer

cosx(3cosx + 2) + 3sin^2x = (2+ 3cosx)* d(sinx)  - sinx * d(2 + 3cosx)


here I am using the fact that derivative of sinx is cosx and of cosx is -sinx

anonymous · Answer

ok so wats the benefit for using this ?

shubhamsrg · Answer

try to compare the whole expression with the quotient rule (of derivation)

anonymous · Answer

ok so it becomes
( sinx )/(2+3cosx)  + C ??

shubhamsrg · Answer

thats right ^_^

anonymous · Answer

WOW TRULY GENIUS !! HATS OFF BRO !

shubhamsrg · Answer

a bit of exaggeration but fine, I'll take it :P
thank you

anonymous · Answer

good one!

anonymous · Answer

One more ??
It's the exclusive integral calculus for Jee main & advanced

shubhamsrg · Answer

not too sure if I'll be able to pull off this one :3