Let T and A be constants, how do I solve the following integral? −A/4 int_0^T sin((2 pi t)/(T)−(4 pi tau)/(T))d tau

Vincent Norman

Vincent Norman

Answered question

2022-10-29

Let T and A be constants, how do I solve the following integral?
A 4 0 T sin ( 2 π t T 4 π τ T ) d τ
The solution of the integral has to be 0

Answer & Explanation

Besagnoe9

Besagnoe9

Beginner2022-10-30Added 9 answers

First notice that 0 2 π sin ( u ) d u = 0 (same if it was cosinus under the integral).
Shifting the variable doesn't change the value:
0 2 π sin ( u + φ ) d u = φ 2 π + φ sin ( u ) d u = φ 0 sin ( u ) d u + 0 2 π sin ( u ) d u + 2 π 2 π + φ sin ( u ) d u = 0 φ sin ( u ) d u + 0 2 π sin ( u ) d u + 0 φ sin ( u + 2 π ) sin ( u ) d u = 0 2 π sin ( u ) d u
And this stays the same if you take the integral over an integer number of periods:
0 2 π sin ( n u + ϕ ) d u = 0 n Z
In the present case φ = 2 π t T is a constant.
And for τ [ 0 , T ] then 4 π τ T [ 0 , 4 π ] covers n=2 times the period.
Therefore your integral is zero without having to do any calculation!

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