Question

Integration of theta with respect to d theta​

Answer 1

I tried to fiddle with a substitution of x=sin(θ)x=sin(θ) and tried to go about the question as I would do with a u-substitution question but I got nowhere. Then that got me thinking: what am I supposed to do to integrate something like this where the variable of the function I'm integrating is different to what I'm integrating with respect to?

Also, another side-question is: if the method for this is not substitution, why? Why isn't there some kind of substitution for this?

I should probably say that whilst trying to find an answer on how to do this, I stumbled upon someone saying that you should take the stuff with theta out in front of the integral and treat it as a constant and then integrate 1 w.r.t. x but I don't know if this is useful in this case, since this is a definite integral.

Answer 2

Answer:I tried to fiddle with a substitution of x=sin(θ)x=sin(θ) and tried to go about the question as I would do with a u-substitution question but I got nowhere. Then that got me thinking: what am I supposed to do to integrate something like this where the variable of the function I'm integrating is different to what I'm integrating with respect to?

Also, another side-question is: if the method for this is not substitution, why? Why isn't there some kind of substitution for this?

I should probably say that whilst trying to find an answer on how to do this, I stumbled upon someone saying that you should take the stuff with theta out in front of the integral and treat it as a constant and then integrate 1 w.r.t. x but I don't know if this is useful in this case, since this is a definite integral.

Explanation: