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∫(x + sinx)/(1 + cosx)dx

∫(x + sinx)/(1 + cosx)dx

Grade:upto college level

1 Answers

RISHI KORDE IIT BHU
17 Points
8 years ago

=∫(x)/(1-cos(x)) dx+sin(x)/(1-cos(x))dx

=I1 +I2

 

I1=∫(x)/(2cos2(x/2)) dx

   =0.5∫(x)*sec2(x/2) dx ...........(use byparts, x as differential and sec2(x/2) as integral)

   =0.5[2*x*tan(x/2)  +  ∫(1)*2*tan(x/2) dx]

   =0.5[2*x*tan(x/2)  +  2*(- 2* ln(|cos(x/2)|) )]

   =xtan(x/2)-2 ln(|cos(x/2)|) +c

 

I2=∫(2*sin(x/2)*cos(x/2))/(2*cos2(x/2)) dx

   =∫tan(x/2) dx

   =-2ln(|cos(x/2)|) + c

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