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

Question

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

kunstgrenwa3 · Accepted Answer

Let   I  =  ∫  x      cos          −      1        ⁡  x  d  xPut       cos          −      1        ⁡  x  =  t  ⇒  x  =  cos  ⁡  t  ⇒  d  x  =  −  sin  ⁡  t  d  t    I  =  ∫  x      cos          −      1        ⁡  x  d  x  =  −  ∫  t  cos  ⁡  t  sin  ⁡  t  d  t    =  −      1    2    ∫  t  ⋅  2  sin  ⁡  t  cos  ⁡  t  d  t  =  −      1    2    ∫  t  ⋅  sin  ⁡  2  t  d  tWhen integrating by parts with t as the first function and sin 2t as the second function, we get  I  =  −      1    2    [  t  ∫  sin  ⁡  2  t  d  t  −  ∫  [      d          d      t        (  t  )  ∫  sin  ⁡  2  t  d  t  ]  d  t    =  −      1    2    [  t            (      −      cos      ⁡      2      t      )        2    +  ∫  1            cos      ⁡      2      t        2    d  t  ]  =      1    4    t  cos  ⁡  2  t  −      1    8    sin  ⁡  2  t  +  C    =      1    4    t  (  2      cos    2    ⁡  t  −  1  )  −      1    8    2  sin  ⁡  t  cos  ⁡  t  +  C    =      1    4    (  2      x    2    −  1  )      cos          −      1        ⁡  x  −      1    4    x      1    −          x      2        +  C

Integrate the function. int x cos^(-1)xdx

Answered question

Answer & Explanation

New Questions in Integral Calculus