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Chapter 7: Problem 64

(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). $$\int \frac{x}{(x+3)^{1 / 5}} d x$$

Short Answer

Expert verified
The integral evaluates to \( \frac{1}{2}(x+3)^2 - 3(x+3) + C \).

Step by step solution

01

Identify the u-substitution

Given the integral \( \int \frac{x}{(x+3)^{1/5}} dx \), we decide on a substitution. Let's set \( u = (x+3)^{1/5} \). This choice simplifies the integrand's denominator.
02

Express dx in terms of du

Differentiate \( u = (x+3)^{1/5} \) to find \( du \). We have \( du = \frac{1}{5}(x+3)^{-4/5} dx \), so \( dx = 5(x+3)^{4/5} du \).
03

Substitute in the integral

Express x in terms of u: \( u^5 = x + 3 \) which gives \( x = u^5 - 3 \). Substitute this into the integral: \( \int \frac{u^5 - 3}{u} \cdot 5u^4 \, du \).
04

Simplify the integral

The integrand becomes \( 5 \int ((u^5 - 3) u^4) du = 5 \int (u^9 - 3u^4) du \). This simplifies the integral for easier evaluation.
05

Evaluate the integral

Use power rule to integrate: \( 5 \left( \frac{u^{10}}{10} - \frac{3u^5}{5} \right) + C = \frac{1}{2}u^{10} - 3u^5 + C \).
06

Substitute back for x

Recall \( u = (x+3)^{1/5} \). Substitute back in to express in terms of x: \( \frac{1}{2}(x+3)^2 - 3(x+3) + C \).
07

Verify using CAS

Use a computer algebra system (CAS) to evaluate \( \int \frac{x}{(x+3)^{1/5}} dx \) and confirm that the result matches \( \frac{1}{2}(x+3)^2 - 3(x+3) + C \).

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Key Concepts

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

Integration Techniques
Integration is a fundamental aspect of calculus and essential for solving many mathematical problems. One of the most powerful techniques used in integration is u-substitution. This technique is especially helpful when dealing with complex algebraic expressions or complicated functions that cannot be easily integrated directly.
  • U-substitution: This method involves substituting a part of the integrand with a new variable, usually denoted as \( u \). The goal is to simplify the integral so that it becomes more manageable.
  • Choosing \( u \): The choice of \( u \) is crucial. Generally, \( u \) is chosen to be a function of \( x \) whose derivative also appears in the integrand, simplifying the differentiation and integration process.
  • Procedure: Once \( u \) is chosen, you find \( du \), in terms of \( dx \), and rewrite the entire integral in terms of \( u \). The rewritten integral should be straightforward to integrate using basic integration rules.
For example, in solving \( \int \frac{x}{(x+3)^{1/5}} \, dx \) using u-substitution makes the process more systematic and easier to handle.
Change of Variables
The change of variables technique is crucial in integration to transform complex integrals into simpler forms. In our example, substituting \( u = (x+3)^{1/5} \) is a classic case of change of variables. This approach not only simplifies the integrand but also transforms the differential component.
  • Change of Variables: This involves expressing both the integrand and the differential in terms of a new variable, \( u \).
  • Transformation: By taking the derivative of \( u \) with respect to \( x \), we can find \( du \), which allows us to replace \( dx \). For instance, if \( du = \frac{1}{5}(x+3)^{-4/5} dx \), then \( dx \) can be expressed as \( 5(x+3)^{4/5} du \).
  • Simplification: The original integral is now completely in terms of \( u \), making it easier to integrate, as seen in Step 4 of our exercise where the integrand is simplified to \( 5 \int (u^9 - 3u^4) du \).
This method of changing variables bridges the gap between complex algebraic expressions and compute-friendly forms.
Calculus Problem Solving
Solving calculus problems often involves a systematic approach, especially when dealing with integrals. Understanding the underlying principles and being methodical in applying them is key.
  • Problem Analysis: Identify the complexity of the integral and select the appropriate technique, such as u-substitution, to simplify the process.
  • Step-by-Step Execution: Follow a logical sequence, from choosing \( u \) and differentiating to substituting and simplifying the integral terms.
  • Verification: Once the integral is solved, it鈥檚 important to verify the solution using tools like a CAS (Computer Algebra System). This ensures that your solution is both accurate and equivalent to the derived form from the integral solving process.
In every calculus problem, building these habits helps in tackling even the complex integrals efficiently, as demonstrated in confirming the equivalency of the solution \( \frac{1}{2}(x+3)^2 - 3(x+3) + C \) using a CAS.

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Most popular questions from this chapter

Transform the given improper integral into a proper integral by making the stated \(u\) -substitution; then approximate the proper integral by Simpson's rule with \(n=10\) subdivisions. Round your answer to three decimal places. $$ \int_{0}^{1} \frac{\cos x}{\sqrt{x}} d x ; u=\sqrt{x} $$

Evaluate the integrals that converge. $$ \int_{1}^{+\infty} \frac{d x}{x \sqrt{x^{2}-1}} $$

For what values of \(p\) does \(\int_{0}^{+\infty} e^{p x} d x\) converge?

For the numerical integration methods of this section, better accuracy of an approximation was obtained by increasing the number of subdivisions of the interval. Another strategy is to use the same number of the subintervals. but to select subintervals of differing lengths. Discuss a scheme for doing this to approximate \(\int_{0}^{4} \sqrt{x} d x\) using a trapezoidal approximation with 4 subintervals. Comment on the advantages and disadvantages of your scheme.

Evaluate the integral. $$ \int \frac{x^{4}+6 x^{3}+10 x^{2}+x}{x^{2}+6 x+10} d x $$
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