Evaluate this integral: a <mrow class="MJX-TeXAtom-ORD"> n </mrow> </msub>

Brice Colon

Brice Colon

Answered question

2022-05-27

Evaluate this integral:
a n = 1 π π π ( T 2 π ) 2 y 2 cos n y   d y

Answer & Explanation

Danube2w

Danube2w

Beginner2022-05-28Added 7 answers

For my own benefit, I need to write out all the steps
Ignoring the constant factor out front, integrating π π y 2 cos ( n y ) d y by parts, we see
π π y 2 cos ( n y ) d y = [ y 2 sin ( n y ) n ] π π 2 n π π y sin ( n y ) d y = [ y 2 sin ( n y ) n ] π π 2 n ( [ y cos ( n y ) n ] π π + 1 n π π cos ( n y ) d y ) = [ y 2 sin ( n y ) n ] π π 2 n ( [ y cos ( n y ) n ] π π + 1 n [ sin ( n y ) n ] π π ) = 1 n 3 ( 2 n 2 π 2 sin ( n π ) + 4 π n cos ( n π ) 2 sin ( n π ) )
so if n is a natural number (so sin ( n π ) = 0, I get
π π y 2 cos ( n y ) d y = 4 π ( 1 ) n n 2 .
Edit: fixed a sign based on the below comment.

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