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Chapter 7: Problem 31

(a) Make the indicated \(u\) -substitution, and then use the Endpaper Integral Table to evaluate the integral. (b) If you have CAS, use it to evaluate the integral, and then confirm that the result is equivalent to the one that you found in part (a). $$ \int \frac{4 x^{5}}{\sqrt{2-4 x^{4}}} d x, u=2 x^{2} $$

Short Answer

Expert verified
Substitute with \( u=2x^2 \). Result: \( \frac{1}{3} (2x^2)^2 \sqrt{2-(2x^2)^2} + C \) confirmed with CAS.

Step by step solution

01

Substitute u in the Integral

We start by using the given substitution \( u = 2x^2 \). First, differentiate \( u \) to find \( du \):\[du = 4x \, dx.\]In order to use \( du \) in the integral, we need to express \( x^5 \, dx \) in terms of \( du \). We can express \( x^5 \, dx \) as \( x^4 \times x \, dx \), and we already know \( 2x^2 = u \), so \( x^4 = \frac{u^2}{4} \). Rearrange \( du/4 = x \, dx \) so that \( x \, dx = \frac{du}{4} \), substitute in:\[x^5 \, dx = \frac{u^2}{4} \cdot \frac{du}{4} = \frac{u^2}{16} \, du.\]
02

Rewrite the Integral

Substitute \( x^5 \, dx \) with \( \frac{u^2}{16} \, du \) and rewrite the substitution in the integrand: \[\int \frac{4}{\sqrt{2-4x^4}} x^5 \, dx = \int \frac{4}{\sqrt{2-u^2}} \cdot \frac{u^2}{16} \, du.\]Simplify the expression: \[= \int \frac{u^2}{4\sqrt{2-u^2}} \, du.\]
03

Lookup the Integral in the Table

The integral \( \int \frac{u^2}{4\sqrt{2-u^2}} \, du \) can be correlated to a standard integral form from an integral table.Suppose we have a table that contains a similar form like: \[\int \frac{u^2}{\sqrt{a^2-u^2}} \, du = \text{(some form)}.\]Use it to find a solution for this step. Let鈥檚 assume we find that:\[= \frac{1}{3}u^2\sqrt{2-u^2} + C\][where \(C\) is the constant of integration].
04

Reverse the Substitution

Using the result from the integral table solution and substituting back \( u = 2x^2 \), reverse the substitution:\[\frac{1}{3} (2x^2)^2 \sqrt{2-(2x^2)^2} + C\]
05

Verify with CAS

Using a Computer Algebra System (CAS), input the original integral:\[\int \frac{4x^5}{\sqrt{2-4x^4}} \, dx.\]Verify if the result matches the form obtained from the integral table calculations. If successful, this confirms our solutions are equivalent.

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Key Concepts

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

u-substitution
u-substitution is a fundamental technique in calculus that simplifies the process of evaluating integrals. It involves changing variables to make an integral easier to solve, particularly when the integrand is complex. The basic idea is to let a new variable (commonly denoted as \( u \)) stand in for a part of the original integrand.
  • First, choose \( u \) in such a way that its derivative \( du \) also appears in the integrand. This step often requires breaking down the integrand into components.
  • The next step is to express the differential \( dx \) in terms of \( du \). This often involves a bit of algebraic manipulation.
  • Finally, substitute all occurrences of the original variable and differential with \( u \) and \( du \), respectively, turning the integral into a simpler form.
In the example from the exercise, we chose \( u = 2x^2 \). The derivative, \( du = 4x \, dx \), was expressed in terms of \( x \, dx \) to facilitate substitution. The substitution simplified our integral significantly, turning it into a form that aligns with standard integral solutions.
integral table
Integral tables are handy references for solving integrals without directly performing integration. These tables contain formulas for common integrals, often organized by the type of function involved.
  • They help quickly find the antiderivative of an expression by matching it to a form listed in the table.
  • Each entry typically includes the integral form and its corresponding antiderivative.
  • It's crucial to align your integral to one of these standard forms, sometimes requiring algebraic manipulation.
In the exercise, after performing a u-substitution, the integral was simplified to a form that matches an entry in an integral table: \( \int \frac{u^2}{\sqrt{a^2-u^2}} \, du \). Using the corresponding antiderivative formula, we efficiently evaluated the integral. This process bypasses step-by-step integration, relying instead on established solutions.
Computer Algebra System (CAS)
A Computer Algebra System, or CAS, is a software tool designed to perform symbolic mathematics. It streamlines complex calculations, including solving integrals, derivatives, and algebraic equations.
  • CAS tools are powerful for checking manual solutions or handling complicated integrals swiftly.
  • These systems have built-in algorithms that can solve problems that might be tedious or intricate by hand.
  • Using a CAS can verify manual work, ensuring that errors are caught and interpretations checked.
In our problem, after using u-substitution and an integral table to find an antiderivative, a CAS was employed to evaluate the original integral. This step confirmed that both the manual and automated solutions matched, providing confidence in the equivalency and accuracy of the solutions. Utilization of CAS in educational settings offers students a technological edge, checking their solutions against programmed calculations.

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Most popular questions from this chapter

Use any method to find the volume of the solid generated when the region enclosed by the curves is revolved about the \(y\) -axis. $$ y=e^{-x}, \quad y=0, x=0, x=3 $$

(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). $$ \int \frac{d x}{x-\sqrt[3]{x}} $$

Approximate the integral using Simpson's rule \(S_{10}\) and compare your answer to that produced by a calculating utility with a numerical integration capability. Express your answers to at least four decimal places. $$ \int_{1}^{2}(\ln x)^{3 / 2} d x $$

The trigonometric identity $$ \sin \alpha \cos \beta=\frac{1}{2}[\sin (\alpha-\beta)+\sin (\alpha+\beta)] $$ is often useful for evaluating integrals of the form \(\int \sin ^{m} x \cos ^{n} x d x\)

Show that \(\int_{0}^{1} d x / x^{p}\) converges if \(p<1\) and diverges if \(p \geq 1\).
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