∫xsin−1⁡xdx is equal to 

xsin1xdx is equal to 

  1. A

    sin1x22x21+14x1x2+C

  2. B

    sin1x42x21+14x1x2+C

  3. C

    sin1x42x21+12x1x2+C

  4. D

    None of the above

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    Solution:

    Let I=x sin1xdx
    On taking sin-1 x as first function and x as second function and integrating by parts, we get
    [ inverse functions comes before,algebraic functions in ILATE]
    I=sin1xxdxddxsin1xxdxdx =sin1xx2211x2x22dx
     =x22sin1x11x21x212dx
    [add and subtract 1 in numerator of second term]
    =x22sin1x1211x2dx+121x21x2dx=x22sin1x12sin1x+121x2dx
     I=x22sin1x12sin1x+1212x1x2+12sin1x+C I=sin1xx2214+14x1x2+C I=sin1x42x21+14x1x2+C
     

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