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Magazine Chapter 11: Problem 7
Evaluate the indefinite integral. $$ \int \frac{d x}{1+\sqrt[3]{x-2}} $$
Short Answer
Expert verified
\( 3(\frac{\sqrt[3]{x - 2}^{2}}{2} - \sqrt[3]{x - 2} + \ln|\sqrt[3]{x - 2} + 1|) + C \)
Step by step solution
01
Substitution
Let \( u = \sqrt[3]{x - 2} \). Then, \( x - 2 = u^3 \). Therefore, \( x = u^3 + 2 \).
02
Differentiate Substitution
Differentiate both sides with respect to \(x\). \( \frac{d(x)}{dx} = 3u^2 \frac{d(u)}{dx} \). Hence, \( dx = 3u^2 du \).
03
Rewrite the Integral
Rewrite the integral in terms of \(u\): \( \int \frac{dx}{1+\sqrt[3]{x - 2}} = \int \frac{3u^2 du}{1 + u} \).
04
Simplify the Integral
Factor out and simplify the integrand: \( 3 \int \frac{u^2}{1 + u} du \).
05
Partial Fraction Decomposition
Divide polynomials to simplify: \( u^2/(1 + u) = u - 1 + \frac{1}{1 + u} \. \) Hence, the integral becomes \( 3 \int (u - 1 + \frac{1}{1+u})du \).
06
Integrate Each Term
Integrate term by term: \( 3 \int (u - 1 + \frac{1}{1+u})du = 3 \(\frac{u^2}{2} - u + \ln|u+1| + C\). \)
07
Substitute Back
Substitute \( u = \sqrt[3]{x - 2} \) back into the integral: \( 3 \(\frac{(\sqrt[3]{x - 2})^2}{2} - \sqrt[3]{x - 2} + \ln|\sqrt[3]{x - 2}+1| + C\). \)
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Integration Techniques
Integration techniques are methods used to find the integral of a function. Integrals are fundamental in calculus, representing areas and accumulated quantities. Different functions require different techniques to integrate them effectively. Some key techniques include:
- Substitution (u-substitution)
- Integration by parts
- Partial fraction decomposition
- Trigonometric identities
u-Substitution
U-substitution is a technique used to simplify the process of integration. It works by making a substitution that simplifies the integrand. Here's how it works:
- Identify a part of the integrand that can be substituted with a single variable, usually represented as u.
- Make the substitution and transform the original integral into a simpler form.
- Integrate with respect to the new variable u.
- Substitute back the original variable to get the final answer.
Partial Fraction Decomposition
Partial fraction decomposition is a technique used to break down complicated fractions into simpler parts that are easier to integrate. The method involves:
- Expressing the given fraction as a sum of simpler fractions.
- Solving for constants by equating coefficients or substituting values.
- Integrating each simpler fraction separately.
Definite and Indefinite Integrals
Integrals can be classified into two types: definite and indefinite. Both are essential in calculus:
- Indefinite Integral: Represents a family of functions and includes a constant of integration (C). It is usually written as \( \int f(x) dx = F(x) + C \).
- Definite Integral: Represents the accumulated quantity or area under a curve between two bounds, providing a numerical value. It is usually written as \( \int_{a}^{b} f(x) dx \).
