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First make a substitution and then use integration by parts to evaluate...

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user6978788 | Student, Undergraduate | (Level 1) Honors

Posted February 5, 2013 at 4:28 AM via web

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First make a substitution and then use integration by parts to evaluate `int_0^pie^(cost)sin2tdt`

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sciencesolve | Teacher | (Level 3) Educator Emeritus

Posted February 5, 2013 at 6:24 AM (Answer #1)

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You need to use the following double angle identity, such that:

`sin 2t =  2 sin t*cos t`

`int_0^pi e^(cos t)*(2 sin t*cos t) dt`

You should come up with the following substitution, such that:

 

`2int_1^(-1) (e^u)*u(-du) = 2int_(-1)^1 (e^u)*u(du)`

You may use integration by parts such that:

`f(u) = u => f'(u) = 1`

`g'(u) = e^udu => g(u) = e^u`

`2int_(-1)^1 (e^u)*u(du) = 2(f(u)*g(u)|_(-1)^1 - int_(-1)^1 f'(u)g(u)du)`

`2int_(-1)^1 (e^u)*u(du) = 2(u*e^u|_(-1)^1 - int_(-1)^1 e^u du)`

`2int_(-1)^1 (e^u)*u(du) = 2e^u(u - 1)|_(-1)^1`

Using the fundamental theorem of calculus yields:

`2int_(-1)^1 (e^u)*u(du) = 2e^1(1 - 1)| - 2e^(-1)(-1 - 1)`

`2int_(-1)^1 (e^u)*u(du) = 4/e`

Hence, evaluating the given definite integral, using substitution and parts, yields `int_0^pi e^(cos t)*(2 sin t*cos t) dt = 4/e.`

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