# Trigonometric Substitution for Integrating Radical Expressions

• silicon_hobo
In summary, Homework Equations state that: -The Attempt at a Solution for the 2nd problem is: \frac{81}{8}\int_0^\frac{\pi}{2}(1-\cos{4\theta})d\theta -For the first problem, the attempt is: \int^3_0\ x^2\sqrt{9-x^2} \ dx -Both problems use trig substitutions and the chain rule.
silicon_hobo

## Homework Statement

Hey, it's me again. This method is giving me some trouble. This is the first problem: $$\int^3_0\ x^2\sqrt{9-x^2} \ dx$$

The second problem is:
$$\int\frac{dx}{\sqrt{2x^2+2x+5}}$$. How do I use a trig. substitution to start on this one?

## The Attempt at a Solution

http://www.mcp-server.com/~lush/shillmud/int2.4a.JPG
I know I need to apply an identity here and then maybe integrate by parts. Also, what's the proper way to transform the limits of integration in this type of substitution? Thank you for your input.

Evaluate for your new limits upon substitution.

$$x=3\sin\theta$$

$$\int_0^3\rightarrow\int_0^\frac{\pi}{2}$$

$$81\int_0^\frac{\pi}{2}\sin^{2}\theta\cos^{2}\theta d\theta$$

$$81\int_0^\frac{\pi}{2}\sin^{2}\theta\cos\theta\cos\theta d\theta$$

$$u=\cos\theta$$
$$du=-\sin\theta d\theta$$

$$dV=\sin^{2}\theta\cos\theta d\theta$$
$$V=\frac 1 3\sin^{3}\theta$$

Use parts and you will notice it is a recursive ... bring your original Integral to the left and all you have to evaluate is $$\int\sin^{2}\theta d\theta$$ which can be simplified using a trig identity $$\sin^{2}\theta=\frac 1 2 (1-\cos{2x})$$

Last edited:
Ok done typing.

For your 2nd problem, complete the square and use a Trig sub! Make sure that your leading term is positive and one.

Okay, I think we agree on the first one:
http://www.mcp-server.com/~lush/shillmud/int2.4a2.JPG
But how do I get rid of that pesky $$d\theta$$?

This is what I've got so far for #2. I'm not sure if I've applied the identity correctly:
http://www.mcp-server.com/~lush/shillmud/int2.4b.JPG

Ah very nice alternative to what I suggested, but don't you love that though ... works both ways! Also, keep in mind what I did ... b/c it becomes very useful to notice the chain rule. What do you mean get rid of d-theta? You evaluated for your new limits, so you don't need to get rid of it.

$$\frac{81}{8}\int_0^\frac{\pi}{2}(1-\cos{4\theta})d\theta$$

#2, you made a mistake when you factored out the 2.

$$\int\frac{dx}{\sqrt{2x^2+2x+5}}$$

$$2x^2+2x+5 \rightarrow 2\left(x^2+x+\frac 5 2\right)$$

You applied everything correctly, now go back and just fix the factoring error.

## What is Trigonometric Substitution?

Trigonometric Substitution is a technique used in calculus to solve integrals involving algebraic expressions and trigonometric functions. It involves substituting trigonometric identities in place of variables in the integral, making it easier to solve.

## When should Trigonometric Substitution be used?

Trigonometric Substitution should be used when an integral involves a combination of square roots and algebraic expressions, and can be simplified using trigonometric identities.

## What are the three main types of Trigonometric Substitutions?

The three main types of Trigonometric Substitutions are:

1. Substitution with sin or cos
2. Substitution with tan or cot
3. Substitution with sec or csc

## What are the common trigonometric identities used in Trigonometric Substitution?

The common trigonometric identities used in Trigonometric Substitution include:

• sin^2x + cos^2x = 1
• 1 + tan^2x = sec^2x
• 1 + cot^2x = csc^2x

## What is the process for solving an integral using Trigonometric Substitution?

The process for solving an integral using Trigonometric Substitution is as follows:

1. Identify the type of integral and choose the appropriate trigonometric substitution.
2. Substitute the variables using the trigonometric identity.
3. Simplify the integral using algebra and trigonometric identities.
4. Solve the resulting integral.
5. Substitute back the original variables to get the final solution.

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