Evaluate the integral using the indicated trigonometric...

Key Concepts

Trigonometric Substitution

Trigonometric substitution is a method used in calculus to simplify integrals containing square roots of quadratic expressions. By substituting a trigonometric function for the variable, the integrals transform such that the radical expressions are simplified using trigonometric identities. This approach is especially useful when the integrals involve terms resembling a^2 + x^2, a^2 - x^2, or x^2 - a^2.

Right Triangle Representation

The right triangle representation is a graphical tool used in trigonometric substitution that helps visualize the relationships between the substituted trigonometric function and the original variable. By constructing a right triangle with sides corresponding to terms in the substitution, one can easily convert between trigonometric expressions and algebraic expressions, which is crucial when returning to the original variable after integration.

Use of Trigonometric Identities

Trigonometric identities, such as the Pythagorean identity, are essential in simplifying the expressions obtained after performing a trigonometric substitution. They allow for canceling out or combining square roots and squared trigonometric functions, thereby reducing the integral to a form that is more straightforward to evaluate.

Substitution Method in Integration

The substitution method is a fundamental strategy in integration that simplifies complicated integrals by transforming them into a more manageable form. In the context of trigonometric substitution, the substitution not only redirects the variable but also leverages known trigonometric relationships to simplify the integral, making the process of finding the antiderivative more systematic.

Transcript

00:01 Okay, because there's a plus.

00:03 We know that this is a tangent substitution.

00:06 And what we want is this x squared to be able to be factored out here.

00:12 So we want this x squared to be nine tangent squared.

00:17 X squared is nine tangent squared theta.

00:21 Or like they said, x is three tangent.

00:24 But that's why we're doing it.

00:26 So then when we put nine minus or nine plus, when we have the square root, of 9 plus x squared, we have the square root of 9 plus 9 tangent squared.

00:38 We can factor out the 9 because 1 plus tangent squared is secant squared.

00:49 That turns into three secant.

00:53 And then x is three tangent.

00:55 And so dx is three secant squared theta, d theta.

01:04 All right.

01:04 So that integral turns into x cubed, which would be 27 tangent cubed.

01:14 Times dx, which would be three secant squared theta, d theta, over that square root, which we calculated to be three secant theta.

01:29 Okay, so the threes cancel, and one of the sequence cancel.

01:33 So we have 27 integral, tangent cube theta, secant theta, d theta.

01:43 Okay, if you have odd of both of them, what you want to do is, take out one of each one so you have a tangent secant.

01:53 So we have tangent squared theta, tangent secant.

02:03 Okay, so we want to make that du, so that means u is the secant squared.