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Magazine Chapter 5: Problem 17
Exer. 9-48: Evaluate the integral. $$ \int \frac{x}{\sqrt[3]{1-2 x^{2}}} d x $$
Short Answer
Expert verified
The integral evaluates to \( -\frac{3}{8}(1-2x^2)^{2/3} + C \).
Step by step solution
01
Identify the Substitution
To evaluate the integral \( \int \frac{x}{\sqrt[3]{1-2x^2}} \, dx \), we notice that the expression \(1-2x^2\) inside the cube root suggests a substitution that will simplify the integration process. Let \( u = 1 - 2x^2 \).
02
Differentiate the Substitution
Differentiate \( u = 1 - 2x^2 \) with respect to \( x \) to find \( du \). This gives \( du = -4x \, dx \), or equivalently, \( dx = \frac{-du}{4x} \).
03
Substitute and Simplify the Integral
Substitute \( u = 1 - 2x^2 \) and \( dx = \frac{-du}{4x} \) into the original integral. The integral becomes \( \int \frac{x}{u^{1/3}} \cdot \frac{-du}{4x} \). Simplify this to \( -\frac{1}{4} \int u^{-1/3} \, du \).
04
Integrate with Respect to \(u\)
The integral \( \int u^{-1/3} \, du \) is a standard power integral. Using the power rule, we have \( \int u^n \, du = \frac{u^{n+1}}{n+1} + C \) for \( n eq -1 \). Here, \( n = -1/3 \), so the integral becomes \( \frac{u^{2/3}}{2/3} + C = \frac{3}{2}u^{2/3} + C \). Thus, our integral becomes \( -\frac{1}{4} \cdot \frac{3}{2}u^{2/3} + C = -\frac{3}{8}u^{2/3} + C \).
05
Substitute Back for \(x\)
Substitute back \( u = 1 - 2x^2 \) into the expression \( -\frac{3}{8}u^{2/3} + C \) to obtain the solution in terms of \( x \). This yields \( -\frac{3}{8}(1-2x^2)^{2/3} + C \).
06
Final Answer
The evaluated integral is \( -\frac{3}{8}(1-2x^2)^{2/3} + C \), where \( C \) is the constant of integration.
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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 integration that simplifies more complex integrals by transforming them into simpler forms. This involves choosing a new variable, often denoted as \( u \), to replace a part of the expression in the integral. Let's break it down:
- Identify the Component for Substitution: In the integral \( \int \frac{x}{\sqrt[3]{1-2x^2}} \, dx \), the term \( 1 - 2x^2 \) is suitable for substitution. By setting \( u = 1 - 2x^2 \), you create a new variable that aids in reducing complexity.
- Differentiate to Find the New Differential: Differentiating \( u \) gives \( du = -4x \, dx \). Solving for \( dx \), we get \( dx = \frac{-du}{4x} \). This transforms the differential in terms of \( x \) into one in terms of \( u \).
- Substitute and Simplify: With \( u \) and \( dx \), substitute back into the integral: \( \int \frac{x}{u^{1/3}} \cdot \frac{-du}{4x} \). Here, the \( x \) terms cancel each other, simplifying it to \(-\frac{1}{4} \int u^{-1/3} \, du\).
Power Rule for Integration
The Power Rule for Integration is a fundamental technique used to integrate expressions in the form of \( x^n \). For integrals of powers, the rule states:
\[ \int x^n \, dx = \frac{x^{n+1}}{n+1} + C \]
This rule is applicable as long as \( n eq -1 \). In our exercise, after substitution, we ended up with the integral \( \int u^{-1/3} \, du \). Here's how the Power Rule works in this context:
\[ \int x^n \, dx = \frac{x^{n+1}}{n+1} + C \]
This rule is applicable as long as \( n eq -1 \). In our exercise, after substitution, we ended up with the integral \( \int u^{-1/3} \, du \). Here's how the Power Rule works in this context:
- Apply the Power Rule: We have \( n = -1/3 \) for \( u \), so according to the rule, the integral becomes \( \frac{u^{2/3}}{2/3} \). Simplifying this gives \( \frac{3}{2}u^{2/3} + C \).
- Integrate and Simplify: Applying the rule efficiently transforms what was once a complicated expression into something manageable.
Definite and Indefinite Integrals
Integrals can be categorized into definite and indefinite, both serving different purposes in calculus.
- Indefinite Integrals: These represent a family of functions and are common when starting with basic anti-differentiation. The presented solution is an indefinite integral, as there are no bounds given and the solution includes a constant \( C \), \( -\frac{3}{8}(1-2x^2)^{2/3} + C \).
- Definite Integrals: These are evaluated over a specific interval and result in a numerical value. They are often used to find areas under curves or between functions.
