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Question

Integrate the following functions w.r.t. x.

1cos(x+a) cos(x+b)dx.

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Solution

1cos(x+a) cos(x+b)dx

On multiplying and dividing by sin(a - b), we get

I=1sin(ab)sin(ab)cos(x+a) cos(x+b)dx I=1sin(ab)sin[(x+a)(x+b)]cos(x+a) cos(x+b)=1sin(ab)sin(x+a)cos(x+b)cos(x+a)sin(x+b)cos(x+a)cos(x+b)dx [ sin(AB)=sin A cos Bsin B cos A]=1sin(ab)[sin(x+a)cos(x+a)sin(x+b)cos(x+b)]dx=1sin(ab)[tan(x+a)tan(x+b)]dx=1sin(ab)[log|cos(x+a)|+log|cos(x+b)|]+C=1sin(ab)log cos(x+b)cos(x+a)+C ( log mlog n=logmn)


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