Help with Trig Substitution Integral Problem

[student93]

Homework Statement

Question is attached in this post.

Homework Equations

Question is attached in this post.

The Attempt at a Solution

I've solved the problem via using x=asinθ where a=1

I've been able to integrate the problem to the point where I get cos^2(θ)/sin^2(θ), but can't seem to find a way to simply the problem further so that I can finish off the integration.

The answer to the question is √3 - π/3

 

[Curious3141]

Homework Statement

Question is attached in this post.

Homework Equations

Question is attached in this post.

The Attempt at a Solution

I've solved the problem via using x=atanθ where a=1

I've been able to integrate the problem to the point where I get cos^2(θ)/sin^2(θ), but can't seem to find a way to simply the problem further so that I can finish off the integration.

The answer to the question is √3 - π/3

Please show your steps in detail.

 

[Tanya Sharma]

I've solved the problem via using x=atanθ where a=1

I've been able to integrate the problem to the point where I get cos^2(θ)/sin^2(θ), but can't seem to find a way to simply the problem further so that I can finish off the integration.

The answer to the question is √3 - π/3

I do not see how x=atanθ would help you in this problem .Instead try x=sinθ or cosθ.

 

[student93]

I do not see how x=atanθ would help you in this problem .Instead try x=sinθ or cosθ.

That was a typo, I actually did use asinθ.

 

[student93]

Please show your steps in detail.

∫√(1-sin^2(θ))/(sin^2(θ) dθ = ∫cos^2(θ)/sin^2(θ) dθ

(I don't know how to simply the problem further, I know cos^2/sin^2=cot^2, but trying to get the integral of cot^2 isn't practical etc.)

 

[Tanya Sharma]

That was a typo, I actually did use asinθ.

Okay...Now rewrite cot²θ in terms of cosec²θ .

 

[student93]

Okay...Now rewrite cot²θ in terms of cosec²θ and proceed .

Thanks lol, I completely forgot about that identity since I haven't used that specific one in a long time etc.