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Question

Solve: tan11x1+xdx

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Solution

Put x=cos2t,
=> dx=2sin2tdt
=> 2tan11cos2t1+cos2tsin2tdt
=> 2tan12sin2t2cos2tsin2tdt
=> 2tan1(tant).sin2tdt
=> 2t.sin2tdt
Solving this using By Parts, we get
=> 2[t2sin2t2+cos2t2]dt
=> [t2sin2t+sin2t2]+C
Put t=cos1x2,
=> [cos1x22.sin(cos1x)+sin(cos1x)2]+C

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