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Question

0asin-1xa+x dx

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Solution

Let, I=0asin-1xa+x dxLet, x=a tan2θ θ=tan-1xaWhen, xx ; θ0 and xa ; θπ4and dx=2a tanθ sec2θ dθThen,I=0π4sin-1a tan2θa+a tan2θ 2a tanθ sec2θ dθI= 2a 0π4sin-1sinθ tanθ sec2θ dθI= 2a 0π4θ tanθ sec2θ dθLet, tan θ=tθ=tan-1tsec2θ dθ =dtwhen, θ0 ; t0 and θπ4 ; t1Then, I=2a01tan-1t t dt =2a01tan-1t t dt =2atan-1t t2201-2a201t21+t2 dt =2aπ4×12-0-a011-11+t2 dt =2aπ8-at-tan-1t01 =πa4-a1-π4 =πa4-a+πa4 =πa2-a =aπ2-1

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