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Question

01cos-11-x21+x2 dx

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Solution

01cos-11-x21+x2dxLet, x= tanθ, dx= sec2θ dθWhen, x0 ; θ0and x1 ; θπ4Therefore, the integral becomes0π4cos-11-tan2θ1+tan2θ sec2θ dθ= 0π4cos-1cos2θ sec2θ dθ=20π4θ sec2θ dθ=2θ tanθ0π4-20π4tanθ dθ=2θ tanθ0π4+2logcosθ0π4=2π4-0+2log12-0=π2-log2

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