Integrate the function. ∫ x cos − 1 ⁡ x 1 − x 2 d x

Question

Integrate the function.  ∫            x              cos                  −          1                    ⁡      x              1      −              x        2              d  x

Marin Valencia · Accepted Answer

Let   I  =  ∫            x              cos                  −          1                    ⁡      x              1      −              x        2              d  x  ⇒  I  =  ∫      cos          −      1        ⁡  x      x          1      −              x        2              d  x Consider       cos          −      1        ⁡  x as first function and       x          1      −              x        2             as a second function and integration the components, we obtain  I  =      cos          −      1        ⁡  x  ∫      x          1      −              x        2              d  x  −  ∫  [      d          d      x        (      cos          −      1        ⁡  x  )  ∫      x          1      −              x        2              d  x  ]  d  x    Let   1  −      x    2    =      t    2    ⇒  −  2  x  =  2  t            d      t              d      x        ⇒  d  x  =  −      t    x    d  t    ∴  I  =      cos          −      1        ⁡  x  ∫      x    t    (            −      t        x    )  d  t  −  ∫  [      d          d      x        (      cos          −      1        ⁡  x  )  ∫      x    t    (            −      t        x    )  d  t  ]  d  x    =      cos          −      1        ⁡  x  (  −  t  )  −  ∫  [            (      −      1      )              1      −              x        2              (  −  t  )  d  t  ]  d  x  [  ∵  1  −      x    2    =      t    2    ⇒  t  =      1    −          x      2        ]    =  −      1    −          x      2        ⋅      cos          −      1        ⁡  x  −  ∫  1  d  x  =  −      1    −          x      2            cos          −      1        ⁡  x  −  x  +  C

Integrate the function. int (x cos^(−1)x)(sqrt(1−x^2))dx.

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