Problem 56
Question
Use the substitution \(u=x^{1 / 6}\) to evaluate \(\int \frac{1}{x^{5 / 6}+x^{2 / 3}} d x\)
Step-by-Step Solution
Verified Answer
The answer is \(6\ln|x^{1/6} + 1|\).
1Step 1: Substitution
Substitute \(u = x^{1/6}\). Thus, \(u^2 = x^{1/3}\) and \(u^3 = x^{1/2}\), leading to \(x^{5/6} = u^5\) and \(x^{2/3} = u^4\). This gives us \(\frac{1}{u^5 + u^4}\). Differentiating our substitution equation, we get \(du = \frac{1}{6}x^{-5/6} dx\), which means \(dx = 6u^5du\).
2Step 2: Replace dx and Simplify
Replacing dx in the integral expression, we get \(\int \frac{6u^5}{u^5 + u^4} du\), which can be simplified as \(\int \frac{6}{1 + u^{-1}} du\).
3Step 3: Integral Evaluation
The integral of this function is 6 times the natural logarithm of \(u + 1\), so \(\int \frac{6}{1 + u^{-1}} du = 6\ln|u + 1|\).
4Step 4: Re-Substitute for x
Re-substituting \(u = x^{1/6}\), we get \(6\ln|x^{1/6} + 1|\).
Key Concepts
Integration TechniquesSubstitution MethodNatural Logarithm
Integration Techniques
Integration is a fundamental concept in calculus, primarily concerned with finding the integral, or the area under a curve. Different functions require different techniques for integration, as not all integrals can be solved using a straightforward approach.
When we talk about integration techniques, we are referring to specific methods used to solve more complex integrals. Some commonly used techniques include:
In our particular problem, we use a technique known as the substitution method, which simplifies solving integrals by changing variables. This is especially useful when the integral includes composite functions, allowing us to turn a complicated integral into a much simpler form. This technique is often employed to handle integrals that involve roots and powers, like in our exercise.
When we talk about integration techniques, we are referring to specific methods used to solve more complex integrals. Some commonly used techniques include:
- The power rule
- Integration by parts
- Partial fraction decomposition
- Substitution method
In our particular problem, we use a technique known as the substitution method, which simplifies solving integrals by changing variables. This is especially useful when the integral includes composite functions, allowing us to turn a complicated integral into a much simpler form. This technique is often employed to handle integrals that involve roots and powers, like in our exercise.
Substitution Method
The substitution method is a powerful tool in integration that involves changing the variable of integration to simplify the problem. The basic idea is to substitute a part of the integral with a new variable, typically denoted as "u." This new variable represents a function of the original variable in the integral.
In our exercise, we make the substitution: - Letting \( u = x^{1/6} \),- Then, \( x^{5/6} \) becomes \( u^5 \) and \( x^{2/3} \) becomes \( u^4 \).These transformations greatly simplify the integrand.
The next step involves differentiating the new variable \( u \) with respect to \( x \) to find an expression for \( dx \). In our exercise, this gives us \( du = \frac{1}{6}x^{-5/6} dx \). Solving for \( dx \) allows us to express all terms in the integral using \( u \) and \( du \).This substitution transforms the original integral into a simpler one, allowing us to integrate with respect to \( u \) instead. Finally, we re-substitute the original variable back into the integral to obtain the solution in terms of the original variable, ensuring the answer directly relates back to the given problem.
In our exercise, we make the substitution: - Letting \( u = x^{1/6} \),- Then, \( x^{5/6} \) becomes \( u^5 \) and \( x^{2/3} \) becomes \( u^4 \).These transformations greatly simplify the integrand.
The next step involves differentiating the new variable \( u \) with respect to \( x \) to find an expression for \( dx \). In our exercise, this gives us \( du = \frac{1}{6}x^{-5/6} dx \). Solving for \( dx \) allows us to express all terms in the integral using \( u \) and \( du \).This substitution transforms the original integral into a simpler one, allowing us to integrate with respect to \( u \) instead. Finally, we re-substitute the original variable back into the integral to obtain the solution in terms of the original variable, ensuring the answer directly relates back to the given problem.
Natural Logarithm
The natural logarithm, denoted as \( \ln \), is a logarithm to the base \( e \), where \( e \approx 2.71828 \) is the Euler's number. The natural logarithm is a very important function in calculus, appearing frequently in various types of integration problems, particularly those involving growth and decay, as well as in exponential functions.
In the context of integration, the natural logarithm often arises when one is dealing with integrals of the form \( \int \frac{1}{x} dx \), which simplifies to \( \ln|x| + C \), where \( C \) is the constant of integration.
In the context of integration, the natural logarithm often arises when one is dealing with integrals of the form \( \int \frac{1}{x} dx \), which simplifies to \( \ln|x| + C \), where \( C \) is the constant of integration.
- For our problem, after substitution and simplification, the integral evaluates to involve a natural logarithm.
- The expression becomes \( 6 \ln|u+1| \), where \( u \) is the substituted variable.
- After integrating in terms of \( u \), you substitute back the original variable \( x \), returning to \( 6\ln|x^{1/6} + 1| \).
Other exercises in this chapter
Problem 55
Find the average value of the function on the given interval. \(f(x)=2 x-2 x^{2},[0,1]\)
View solution Problem 55
Use a geometric formula to compute the integral. $$\int_{0}^{2} 3 x d x$$
View solution Problem 56
Find the average value of the function on the given interval. \(f(x)=x^{3}-3 x^{2}+2 x,[1,2]\)
View solution Problem 56
Use a geometric formula to compute the integral. $$\int_{1}^{4} 2 x d x$$
View solution