1. I used the identity
I tried letting . So
.
I need an explanation from top to bottom guys, please.
I'm purposefully trying not to use partial fraction decomp.
Hello, VonNemo19!
There is a formula for this problemL . .
No one has taught it to you?
You can derive the formula like this . . .
We have: .
Multiply by .
Let
. . Hence: .
Substitute: .
Back-substitute: .
Multiply by .
Since , we have: .
Now apply this formula to your problem . . .
Soroban's identity is elegant, and standard. It's good to know the integration formula for every trigonometric function. More importantly, the solution you derived was for a different integral, and it only (directly) involves the sine function because there was a double angle involved. In other words, he was suggesting you derive your result from the formula for
whereas you derived from scatch a formula for
which is somewhat different.
Incidentally, and apropos of nothing, do you know the method of Weierstrass substitution? It transforms any rational function in terms of sine and cosine into an ordinary rational function in a single variable.