Spotting substitution in ∫ 1 ( t + cos ⁡ α ) 2 + sin 2 ⁡ α

Question

Spotting substitution in   ∫      1          (      t      +      cos      ⁡      α              )        2            +              sin        2            ⁡      α

Julien Carrillo · Accepted Answer

HINT:  ∫      1                            (          t          +          cos          ⁡          (          a          )          )                2            +              sin        2            ⁡      (      a      )           d  t  =Substitute   u  =  cos  ⁡  (  a  )  +  t and du=dt:  ∫      1                  sin        2            ⁡      (      a      )      +              u        2                 d  u  =  ∫                    csc        2            ⁡      (      a      )                      u        2                    csc        2            ⁡      (      a      )      +      1           d  u  =      csc    2    ⁡  (  a  )  ∫      1                  u        2                    csc        2            ⁡      (      a      )      +      1           d  u  =Substitute s=ucsc(a) and ds=csc(a) du:  csc  ⁡  (  a  )  ∫      1                  s        2            +      1           d  s

Trevor Wood · Accepted Answer

Factoring gives      1                  sin        2            ⁡      α        ∫            d      t              (      t      csc      ⁡      α      +      cot      ⁡      α              )        2            +      1        ,so that the denominator of the integral has the familiar form       v    2    +  1. This suggests writing  t  csc  ⁡  α  +  cot  ⁡  α  =  v  =  tan  ⁡  u  ,and multiplying both sides of this substitution by sinα gives the desired substitution:                                          sin            ⁡            α            tan            ⁡            u            =            t            +            cos            ⁡            α                                .

Spotting substitution in ∫ 1 ( t + c

Answered question

Answer & Explanation

New Questions in Precalculus