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Integrate: (x + sin x)/(1+cos x)

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Solution: We have ,

                 I=int fracx+sin x1+cos xhspace0.2cmdx

        \ Rightarrow I=int fracx+2sin fracx2cdot cos fracx22cos^2(fracx2)hspace0.2cmdx\ \Rightarrow I=int [frac12xsec^2(fracx2)+ an (fracx2)]hspace0.2cmdx

        Put ,   fracx2=tRightarrow (frac12)dx=dt

       \Rightarrow I= 2int [tsec^2t+ an t] dt

     herefore     I=2[t an t-int an t hspace0.2cmdt+int an thspace0.2cmdt]\ \ Rightarrow I=2t an t+c=x an (fracx2)+c.

Posted by

Deependra Verma

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