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Integrate:(1+xcosx)/x(1-x^2e^2sinx)

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Solution: Let ,

$I=int (1+cosx)/x(1-x^2e^2sinx)hspace0.1cmdx$

put, $xe^sinx=tRightarrow xe^sinxcdot cosx+e^sinxhspace0.1cmdx=dt$

$e^sinx(xcosx+1)dx=dt$

$\Rightarrow I=int 1+xcosx/x(1-x^2e^2sinx)hspace0.1cmdx$

$I=int 1/t(1-t^2)hspace0.1cmdt=int 1/t(1-t)(1+t)hspace0.1cmdt\ \Rightarrow I=[1/t+1/2(1-t)-1/2(1+t)]dt\ \Rightarrow I=log left | t ight |-1/2log left | 1-t ight |-1/2log left | 1+t ight |+c\ \Rightarrow I=log left | xe^sinx ight |-1/2log left | 1-x^2e^2sinx ight |+c$

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Deependra Verma

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