# Help with Trig Substitution Integral Problem

• student93
In summary: Thanks lol, I completely forgot about that identity since I haven't used that specific one in a long time.
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

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student93 said:

## 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

student93 said:
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.

Tanya Sharma said:
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.

Curious3141 said:

∫√(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.)

student93 said:
That was a typo, I actually did use asinθ.

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

Tanya Sharma said:
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.

## 1. What is trig substitution and when should it be used?

Trig substitution is a technique used to solve integrals that involve expressions containing square roots of quadratic polynomials. It involves replacing the variable in the integral with a trigonometric function. This technique should be used when the integral cannot be solved using other methods like u-substitution or integration by parts.

## 2. How do I know which trig function to use in trig substitution?

The choice of trigonometric function depends on the expression inside the square root in the integral. If the expression is of the form a2 - x2, use sine substitution; if the expression is of the form x2 - a2, use cosine substitution; and if the expression is of the form x2 + a2, use tangent substitution.

## 3. Can trig substitution be used for all types of integrals?

No, trig substitution can only be used for integrals that involve expressions containing square roots of quadratic polynomials. For other types of integrals, other techniques like u-substitution, integration by parts, or partial fraction decomposition should be used.

## 4. How do I determine the limits of integration when using trig substitution?

When using trig substitution, the limits of integration should also be changed to match the new variable. This can be done by substituting the original limits into the trigonometric function and simplifying the resulting expression. It is important to keep track of the changes to the limits during the substitution process.

## 5. Are there any common mistakes to avoid when using trig substitution?

One common mistake to avoid is forgetting to change the limits of integration when using trig substitution. Another mistake is using the wrong trigonometric function, which can result in an incorrect solution. It is also important to carefully simplify the resulting expression after substitution to avoid errors.

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