Question

integrate cosx/1+tanx
please solve ​

Answer 1

Answer:

hii here is ur answer!!

*A2A*

cos2x1+tanx=cos3xcosx+sinx  

Let  I=∫cos3xcosx+sinxdx  and  J=∫sin3xcosx+sinxdx  

I+J=∫cos3x+sin3xcosx+sinxdx=∫(1−sinxcosx)dx=x+12cos2x+C1  

I−J=∫cos3x−sin3xcosx+sinxdx=∫(cosx−sinx)(1+sinxcosx)cosx+sinxdx  

Put  cosx+sinx=t  to get,

I−J=12∫t2+1tdt=12∫t+1tdt=14t2+12logt+C2  

=14(cosx+sinx)2+12log(cosx+sinx)+C2  

Thus,

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Now,  I=I+J+I−J2  

Therefore,

I=x2+14cos2x+18(cosx+sinx)2+14log(cosx+sinx)+C

hope helpful!