1)Find the indefinite integral. (Use C for the constant of integration.)int frac{cos^5 t}{sqrt{sin t}} dt2)Use Wallis's Formulas to evaluate the integral.int_0^{frac{pi}{2}} sin^7 x dx

Question

1)Find the indefinite integral. (Use C for the constant of integration.)int frac{cos^5 t}{sqrt{sin t}} dt2)Use Wallis&#039;s Formulas to evaluate the integral.int_0^{frac{pi}{2}} sin^7 x dx

Micah Hobbs · Accepted Answer

1)  ∫                    cos        5            ⁡      t              sin      ⁡      t        d  tput   sin  ⁡  t  =  u  ⇒  cos  ⁡  t  d  t  =  d  u  ∫                    cos        5            ⁡      t              sin      ⁡      t        d  t  =  ∫                    cos        4            ⁡      t      cos      ⁡      t              sin      ⁡      t        d  t  =  ∫            (              cos        2            ⁡      t              )        2            cos      ⁡      t              sin      ⁡      t        d  t  =  ∫            (      1      −              sin        2            ⁡      t              )        2            cos      ⁡      t              sin      ⁡      t        d  t  =  ∫            (      1      −              u        2                    )        2                    u        d  u  =  ∫            (      1      +              u        4            −      2              u        2            )              u        d  u  =  ∫      1          u        d  u  +  ∫            u      4              u        d  u  −  2  ∫            u      2              u        d  u  =  ∫      u                            −          1                2              d  u  +  ∫      u          4      −              1        2              d  u  −  2  ∫      u          2      −              1        2              d  u  =  ∫      u          −              1        2              d  u  +  ∫      u                  7        2              d  u  −  2  ∫      u                  3        2              d  u  =            u                                    −            1                    2                +        1                    1      2        +            u                        7          2                +        1                    9      2        −  2            u                        3          2                +        1                    5      2        +  C  =  2      u                  1        2              +      2    9        u                  9        2              −  2  (      2    5        u                  5        2              )  +  C  =  2      u    +      2    9        u                  9        2              −      4    5        u                  5        2              +  C  =  2      sin    ⁡    t    +      2    9        sin                  9        2              ⁡  t  −      4    5        sin                  5        2              ⁡  t  +  C

Keenan Conway · Accepted Answer

∫    0                  π        2                  sin    7    ⁡  x  d  x  =            (      7      −      1      )      (      7      −      3      )      (      7      −      5      )              7      (      7      −      2      )      (      7      −      4      )        =            6      (      4      )      (      2      )              7      (      5      )      (      3      )        =            2      (      4      )      (      2      )              7      (      5      )        =      16    35

1)Find the indefinite integral. (Use C for the constant of integration.) int (cos^5 t)/(sqrt(sin t)) dt 2)Use Wallis's Formulas to evaluate the integral. int_0^((pi)/(2)) sin^7 x dx

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