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Chapter 8: Problem 39

Trigonometric substitutions Evaluate the following integrals using trigonometric substitution. $$\int \frac{x^{2}}{\left(100-x^{2}\right)^{3 / 2}} d x$$

Short Answer

Expert verified
Question: Evaluate the integral $$\int \frac{x^{2}}{\left(100-x^{2}\right)^{3 / 2}} d x$$ using trigonometric substitution. Answer: $$\int \frac{x^{2}}{\left(100-x^{2}\right)^{3 / 2}} d x = -\frac{x}{\sqrt{100 - x^2}} + 5\ln\left|\frac{x + \sqrt{100 - x^2}}{x - \sqrt{100 - x^2}}\right| + C$$

Step by step solution

01

Choose the trigonometric substitution

Let's choose the substitution: $$x = 10\sin{u}$$ Then, differentiating with respect to u, we get: $$dx = 10\cos{u} du$$
02

Rewrite the integral using the substitution

Substitute \(x\) and \(dx\) in the integral, we have: $$\int \frac{(10\sin{u})^{2}}{\left(100 - (10\sin{u})^{2}\right)^{3 / 2}} (10\cos{u})du$$
03

Simplify the integral

Let's simplify the integral: $$\int \frac{100\sin^{2}{u}}{\left(100 - 100\sin^{2}{u}\right)^{3 / 2}} 10\cos{u} du$$ Now, recall the identity $$\cos^2{u} = 1 - \sin^2{u}$$. We can rewrite the integral as: $$\int \frac{100\sin^{2}{u}}{\left(100(1-\sin^{2}{u})\right)^{3 / 2}} 10\cos{u} du$$ Further simplifying, we get: $$\int \frac{10\sin^{2}{u}\cos{u}}{(1-\sin^{2}{u})^{\frac{3}{2}}} du$$
04

Perform the integral

Let's now perform the integral using substitution. Let \(v = \sin{u}\), then \(dv = \cos{u} du\). Our integral now becomes: $$\int \frac{10v^{2}}{(1-v^{2})^{\frac{3}{2}}} dv$$ This integral can now be solved using standard integration techniques such as partial fraction decomposition or direct integration. $$\int \frac{10v^2}{(1-v^2)^{\frac{3}{2}}} dv = -\frac{10v}{\sqrt{1 - v^2}} + \frac{10}{2}\ln|\frac{v + \sqrt{1-v^2}}{v -\sqrt{1-v^2}}|+ C$$
05

Substitute back for u and x

Now, let's substitute back for \(u\) and \(x\): $$-\frac{10\sin{u}}{\sqrt{1 - \sin^2{u}}} + \frac{10}{2}\ln|\frac{\sin{u} + \sqrt{1-\sin^2{u}}}{\sin{u} -\sqrt{1-\sin^2{u}}}|+ C$$ Then, substituting for x: $$-\frac{10(10\sin{u})}{10\cos{u}} + \frac{10}{2}\ln|\frac{10\sin{u} + 10\cos{u}}{10\sin{u} - 10\cos{u}}|+ C$$ Finally, we simplify: $$-\frac{x}{\sqrt{100 - x^2}} + 5\ln\left|\frac{x + \sqrt{100 - x^2}}{x - \sqrt{100 - x^2}}\right| + C$$ Our final answer for the integral is: $$\int \frac{x^{2}}{\left(100-x^{2}\right)^{3 / 2}} d x = -\frac{x}{\sqrt{100 - x^2}} + 5\ln\left|\frac{x + \sqrt{100 - x^2}}{x - \sqrt{100 - x^2}}\right| + C$$

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Integral Calculus
Integral calculus is a branch of mathematics focused on finding the antiderivatives of functions, which are commonly represented as integrals. It provides a way to calculate areas, volumes, and the total accumulation of quantities that vary continuously. Understanding integrals requires knowledge of various integration techniques, one of which is trigonometric substitution. This comes in handy when dealing with integrals containing square roots or other expressions that are difficult to integrate using standard methods. Trigonometric substitution involves using trigonometric identities to simplify an integrand that contains radical expressions with quadratic polynomials.

For example, when faced with an integral such as \(\frac{x^{2}}{(100-x^{2})^{3 / 2}}\), the presence of the square root in the denominator makes direct integration challenging. Utilizing the substitution technique allows us to replace variables with trigonometric functions to simplify the integrand and make the integral more approachable. As demonstrated in the original exercise's solution, choosing \(x = 10\sin{u}\) simplifies the expression significantly, leading us towards a solution.
Trigonometric Identities
Trigonometric identities are equalities involving trigonometric functions that are true for all values of the involved variables. A solid understanding of these identities is crucial for solving calculus problems, particularly when performing trigonometric substitutions in integration. Some of the most fundamental identities include the Pythagorean identities, such as \(\sin^2{u} + \cos^2{u} = 1\), which are used to simplify expressions involving sine and cosine.

In the context of trigonometric substitution for integrals, identities enable us to transform complicated equations into simpler forms that are more manageable to integrate. For example, in the step-by-step solution provided, we employed the identity \(\cos^2{u} = 1 - \sin^2{u}\) to rewrite the integrand. This was a pivotal step that allowed further simplification and eventual integration, demonstrating the power and utility of trigonometric identities in integral calculus.
Integration Techniques
Integration techniques are various methods used to evaluate integrals, which are essential when an antiderivative cannot be easily found. These techniques include, but are not limited to, substitution, integration by parts, partial fractions, and trigonometric substitution, as shown in the exercise. Trigonometric substitution is particularly useful when the integrand involves a square root of a quadratic expression or when the denominator can be expressed in terms of sine and cosine.

Once a trigonometric substitution is made, as in our example with the substitution \(x = 10\sin{u}\), the integral often simplifies to a form that can be integrated directly or using another technique, such as partial fractions. After performing the integral, as in the final steps of the provided solution, it is important to back-substitute the trigonometric function's variable with the original variable to complete the integration process and find the integral in terms of the original variable.

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