How to evaluate &#x222B;<!-- ∫ --> x n </msup> e <mrow class="MJX-TeXAtom

tinydancer27br

tinydancer27br

Answered question

2022-05-22

How to evaluate x n e x cos x d x and x n e x sin x d x ?

Answer & Explanation

Jarrett Reyes

Jarrett Reyes

Beginner2022-05-23Added 8 answers

Note that
x n e a x d x = d n d a n e a x d x = d n d a n ( e a x a ) = k = 0 n ( 1 ) k n ! ( n k ) ! x n k e a x a k + 1
Then
x n e x cos x d x = x n e ( 1 + i ) x d x = k = 0 n ( 1 ) k n ! ( n k ) ! x n k e ( 1 + i ) x ( 1 + i ) k + 1 = k = 0 n ( 1 ) k n ! x n k e x 2 k + 1 2 ( n k ) ! cos ( x ( k + 1 ) π 4 ) x n e x sin x d x = k = 0 n ( 1 ) k n ! x n k e x 2 k + 1 2 ( n k ) ! sin ( x ( k + 1 ) π 4 )

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