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Chapter 7: Problem 10

Evaluate the integral. $$ \int x^{3} \sqrt{5-x^{2}} d x $$

Short Answer

Expert verified
\(-\frac{5}{3}(5-x^2)^{3/2} + \frac{1}{5}(5-x^2)^{5/2} + C\).

Step by step solution

01

Substitution Method

Begin by making a substitution to simplify the integral. Let \( u = 5 - x^2 \). Then, differentiate to find \( du \). We have \( du = -2x \, dx \), or \( x \, dx = -\frac{1}{2} du \). Our integral becomes \(-\frac{1}{2} \int x^2 \sqrt{u} \, du \).
02

Express \( x^2 \) in Terms of \( u \)

From the substitution \( u = 5 - x^2 \), we have \( x^2 = 5 - u \). Substitute this into the integral: \(-\frac{1}{2} \int (5-u)\sqrt{u} \, du \).
03

Split the Integral

Split the integral into two separate integrals: \(-\frac{1}{2} ( \int 5\sqrt{u} \, du - \int u\sqrt{u} \, du )\). Simplifying further, we get \(-\frac{1}{2} ( 5 \int u^{1/2} \, du - \int u^{3/2} \, du )\).
04

Integrate Term by Term

Integrate each term separately: \( \int u^{1/2} \, du = \frac{2}{3}u^{3/2} + C_1 \) and \( \int u^{3/2} \, du = \frac{2}{5}u^{5/2} + C_2 \). The entire integral is: \(-\frac{1}{2} \left( 5\left(\frac{2}{3}u^{3/2}\right) - \left(\frac{2}{5}u^{5/2}\right) \right)\).
05

Simplify the Integral

Simplify the expression: \(-\frac{1}{2} \left( \frac{10}{3}u^{3/2} - \frac{2}{5}u^{5/2} \right) \). So, the integral becomes \(-\frac{5}{3}u^{3/2} + \frac{1}{5}u^{5/2}\).
06

Back-Substitute \( u \) for \( 5 - x^2 \)

Replace \( u \) back with \( 5 - x^2 \) to obtain the expression in terms of \( x \): \(-\frac{5}{3}(5-x^2)^{3/2} + \frac{1}{5}(5-x^2)^{5/2} + C \).
07

Final Simplification

The solution is now written in its most simplified form. You may choose to multiply through by factors received during integration and simplify carefully. The solution remains: \(-\frac{5}{3}(5-x^2)^{3/2} + \frac{1}{5}(5-x^2)^{5/2} + C \).

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Key Concepts

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

Substitution Method
The substitution method is a powerful technique in integral calculus used to simplify the process of finding integrals, especially when the integral involves composite functions. The idea is to transform a complicated integral into a simpler one by substituting part of the integral with a new variable. Typically, this involves identifying a part of the integrand that can be expressed as a derivative.

To perform this method, you follow these steps:
  • First, choose a substitution, say, let \( u = g(x) \). This choice should simplify the integral significantly.
  • Next, differentiate the substitution to find \( du \). Remember that if \( u = g(x) \), then \( du = g'(x) \, dx \).
  • Replace the identified section and its differential \( dx \) in the original integral with \( u \) and \( du \), respectively.
In our exercise, by letting \( u = 5-x^2 \), we express \( x \, dx \) in terms of \( du \) to simplify the integral, which allows us to tackle the problem more efficiently.
Definite and Indefinite Integrals
Integrals are a fundamental component of calculus, and they come in two primary types: definite and indefinite integrals. Understanding the difference is crucial for mastering calculus:

  • **Indefinite Integrals:** These represent antiderivatives of functions. The result is a family of functions plus a constant of integration, \( C \), that signifies an entire range of antiderivatives. A standard example is \( \int f(x) \, dx = F(x) + C \), where \( F'(x) = f(x) \).
  • **Definite Integrals:** These provide a specific number, representing the area under the curve of the function between two points. Denoted as \( \int_{a}^{b} f(x) \, dx \), it calculates the total signed area from \( x = a \) to \( x = b \).
In our exercise, the integral \( \int x^{3} \sqrt{5-x^{2}} \, dx \) is evaluated as an indefinite integral, focusing on finding the general antiderivative without specific limits. This method allows us to find a function that, when differentiated, returns the original integrand.
Integration Techniques
Beyond the substitution method, several integration techniques are employed to solve complex integrals. These strategies can transform a challenging problem into a manageable one by applying appropriate methods:

  • **Integration by Parts:** This technique applies to integrands that are a product of two functions. Based on the product rule for differentiation, it's expressed as \( \int u \, dv = uv - \int v \, du \).
  • **Trigonometric Substitution:** Especially useful for integrals involving square roots of quadratic expressions, where you replace variables using trigonometric identities.
  • **Partial Fraction Decomposition:** Utilized when the integrand is a rational function, it involves breaking down complex expressions into simpler fractions that are easier to integrate.
In this exercise, the substitution method was the chosen strategy due to its effectiveness in handling the given problem. By converting variables, the integral becomes simpler to solve, showcasing the utility of picking the right technique for the task at hand. As you practice and explore various integrations, familiarity with these techniques will enhance your problem-solving skills.

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Most popular questions from this chapter

To evaluate \(\int \sin ^{5} x \cos ^{8} x d x\), use the trigonometric identity \(\sin ^{2} x=1-\cos ^{2} x\) and the substitution \(u=\cos x\).

Use Simpson's rule approximation \(S_{10}\) to approximate the length of the curve over the stated interval. Express your answers to at least four decimal places. y=x^{-2} \text { from } x=1 \text { to } x=2

To evaluate \(\int \sin ^{8} x \cos ^{5} x d x\), use the trigonometric identity \(\sin ^{2} x=1-\cos ^{2} x\) and the substitution \(u=\cos x\).

Numerical integration methods can be used in problems where only measured or experimentally determined values of the integrand are available. Use Simpson's rule to estimate the value of the relevant integral in these exercises. The accompanying table gives the speeds of a bullet at various distances from the muzzle of a rifle. Use these values to approximate the number of seconds for the bullet to travel \(1800 \mathrm{ft}\). Express your answer to the nearest hundredth of a second. [Hint: If \(v\) is the speed of the bullet and \(x\) is the distance traveled, then \(v=d x / d t\) so that \(d t / d x=1 / v\) and \(\left.t=\int_{0}^{1800}(1 / v) d x .\right]\) $$ \begin{array}{rc} \hline {\text { DISTANCE } x \text { (ft) }} & \text { SPEED } v \text { (ft/s) } \\ \hline 0 & 3100 \\ 300 & 2908 \\ 600 & 2725 \\ 900 & 2549 \\ 1200 & 2379 \\ 1500 & 2216 \\ 1800 & 2059 \\ \hline \end{array} $$

(a) Make an appropriate \(u\) -substitution of the form \(u=x^{1 / n}\) or \(u=(x+a)^{1 / n}\), and then evaluate the integral. (b) If you have a CAS, use it to evaluate the integral, and then confirm that the result is equivalent to the one that you found in part (a). $$ \int \frac{d x}{x\left(1-x^{1 / 4}\right)} $$
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