In Exercises 1-4, evaluate the integral by following the...

In Exercises $1-4,$ evaluate the integral by following the steps given. $$I=\int \frac{d x}{x^{2} \sqrt{x^{2}-2}}$$ \begin{equation}\begin{array}{l}{\text { (a) Show that the substitution } x=\sqrt{2} \sec \theta \text { transforms the integral } I} \\ {\text { into } \frac{1}{2} \int \cos \theta d \theta, \text { and evaluate } I \text { in terms of } \theta \text { . }} \\ {\text { (b) Use a right triangle to show that with the above substitution, }} \\ {\sin \theta=\sqrt{x^{2}-2 / x}} \\ {\text { (c) Evaluate } I \text { in terms of } x \text { . }}\end{array}\end{equation}

In Exercises $1-4,$ evaluate the integral by following the steps given.
$$I=\int \frac{d x}{x^{2} \sqrt{x^{2}-2}}$$
\begin{equation}\begin{array}{l}{\text { (a) Show that the substitution } x=\sqrt{2} \sec \theta \text { transforms the integral } I} \\ {\text { into } \frac{1}{2} \int \cos \theta d \theta, \text { and evaluate } I \text { in terms of } \theta \text { . }} \\ {\text { (b) Use a right triangle to show that with the above substitution, }} \\ {\sin \theta=\sqrt{x^{2}-2 / x}} \\ {\text { (c) Evaluate } I \text { in terms of } x \text { . }}\end{array}\end{equation}

Key Concepts

Simplification using Trigonometric Identities

Trigonometric identities, such as the ones involving sine, cosine, and secant functions, are used to simplify integrals after a trigonometric substitution. These identities enable the replacement or combination of trigonometric expressions in the integrand, streamlining the integration process and aiding in the conversion back to the original variables.

Integration by Substitution

Integration by substitution is a method that involves replacing a complicated expression within an integrand with a new variable. This technique, which is essentially the reverse application of the chain rule from differentiation, simplifies the integral so that standard integration methods can be applied more easily.

Trigonometric Substitution

Trigonometric substitution is a technique used to simplify integrals involving square roots of quadratic expressions. By substituting a trigonometric function for the variable, the square root expression can be transformed using fundamental trigonometric identities, such as the Pythagorean identities, thereby reducing the integral to a form that is easier to evaluate.

Right Triangle Relationships

Right triangle relationships provide a geometric means to relate trigonometric functions to algebraic expressions. When using a trigonometric substitution, constructing a right triangle helps in expressing trigonometric functions in terms of the original variable, which is crucial for reverting back to the initial variable after integration.

Transcript

00:01 Let's say i wanted to evaluate the integral i of dx over x squared times root x squared minus 2, using the trigonometric substitution of x equals root 2, secant, theta.

00:18 So what happens if we are to apply this substitution? well, so we're going to want to plug in everywhere we see x.

00:25 We're going to want to plug in root 2, secant theta.

00:28 So this integral is going to become, so we'll do the dx in a second, but it's going to be something over x squared, which is two secant squared theta, two secant squared theta, times the square root, the square root of x squared minus two.

00:53 So x squared is once again this two secant squared theta, two secant squared, theta minus two.

01:08 And this dx, well dx, if we take the derivative of both sides here, dx is going to equal root two times the derivative of secant is secant times tangent.

01:22 So on this side, we're going to have secant theta tan theta d theta.

01:31 And so that's what we're going to replace dx with here.

01:33 So on the top, we have root two, secant, theta, tan theta, d theta.

01:47 So now we want to see what happens here under this radical sign.

01:53 So this root two, we can pull out a root two here.

01:59 And so focusing just on this bottom part, this part, so we have the two secant squared data still.

02:07 And then now we have a times root two times, times the square root of secant squared minus one, which is a trick identity.

02:17 Secant squared minus one is actually just tan squared of theta.

02:24 So we can simplify this root tan squared data to just tan theta, which is great.

02:33 So now our integral will become the integral of root two secant...