Find the integral of <msubsup> &#x222B;<!-- ∫ --> 0 <mrow class="MJX-TeXAtom-OR

Mohammad Cannon

Mohammad Cannon

Answered question

2022-06-21

Find the integral of
0 π 2 sin ( 2 θ ) sin ( θ ) d θ
I got
I = 0 π 4 sin ( 2 θ ) ( sin ( θ ) + cos ( θ ) ) d θ
But, I'm stuck here.

Answer & Explanation

britspears523jp

britspears523jp

Beginner2022-06-22Added 28 answers

If you are interested in knowing how to do this without using the Beta function, try the following steps. But I'm not going to write it out in full because it would take too long.
Call the integral I. First do integration by parts, and we find that
I = 0 π 2 cos 2 θ cos θ sin 2 θ d θ
Now add this form of I to the original form and get
2 I = 0 π 2 cos θ sin 2 θ d θ
hence
4 I = 0 π 2 sin 2 θ sin θ d θ
Now substitute t = tan θ and you end up with a well known integral, featured many times on MSE, requiring a rather tedious partial fraction decomposition, but you get there in the end...
I hope this is sufficient.
Abram Boyd

Abram Boyd

Beginner2022-06-23Added 5 answers

The integral
I = 0 π / 2 sin ( 2 θ ) sin θ d θ
is evaluated by making use of the Beta function. This is seen as follows.
I = 0 π / 2 sin ( 2 θ ) sin θ d θ = 2 0 π / 2 sin 3 / 2 ( θ ) cos 1 / 2 ( θ ) d θ = 2 1 2 B ( 3 4 , 1 4 ) = Γ ( 1 4 ) Γ ( 3 4 ) 4 2 = π 4 .

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