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Chapter 7: Problem 90

Some integrals that can be evaluated by hand cannot be evaluated by all computer algebra systems. Evaluate the integral by hand, and determine if it can be evaluated on your CAS. $$ \int \frac{x^{3}}{\sqrt{1-x^{8}}} d x $$

Short Answer

Expert verified
The integral can be evaluated by hand using a substitution and is complex for some CAS to handle.

Step by step solution

01

Set Up the Substitution

To evaluate the integral \( \int \frac{x^3}{\sqrt{1-x^8}} \, dx \), we start with a substitution. Let \( u = 1 - x^8 \), then we have \( du = -8x^7 \, dx \). However, to adjust for \( x^3 \, dx \), note that \( x^4 = (1-u)^{1/8} \), so \( x^3 \, dx \) becomes \( \frac{1}{8} \cdot \frac{du}{-x^4} = -\frac{1}{8\sqrt{x^8}} du \).
02

Change the Variables in the Integral

Using the substitution \( x^8 = 1-u \), we have \( \sqrt{1-x^8} = \sqrt{u} \). Additionally, express \( x^3 = (1-u)^{3/8} \). Substitute these into the integral, so it becomes \( -\frac{1}{8}\int \frac{(1-u)^{3/8}}{\sqrt{u}\sqrt{1-u}} \, du \). Simplify further if necessary.
03

Simplify and Evaluate the Integral

Since the terms are complex, you can split and simplify further if the form permits factorization or use partial fraction decomposition. For instance, rewrite \( (1-u)^{3/8} \) if possible for easier handling. Factors or known identities might be used for final evaluation.
04

Determine CAS Compatibility

Attempt to evaluate the integral using a Computer Algebra System (CAS), such as Wolfram Alpha, Maple, or Mathematica. If the CAS can handle complex substitutions and transformations effectively, it should provide a solution. Evaluate whether the CAS gives a direct result or indicates complexity in solving.

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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 one of the simplest yet most powerful techniques for evaluating integrals, especially when the function you're dealing with is complex. This method involves changing the variable of integration to something simpler, which can help transform a complex integral into one that is easier to evaluate. Here's how it typically works:
  • First, identify a substitution variable, often designated as \( u \), that can simplify the integral. For instance, in the exercise, we let \( u = 1 - x^8 \).
  • Next, calculate the differential \( du \). For our exercise, starting from \( u = 1 - x^8 \), we have \( du = -8x^7 \, dx \).
  • Replace all instances of the original variable and its differential in the integral with the terms related to \( du \). This is how the original integral is transformed into an integral in terms of \( u \): it becomes less daunting and more approachable.
  • Finally, integrate with respect to \( u \), and don鈥檛 forget to change back to the original variable if needed after finding the antiderivative.
This technique is especially useful when the expression contains forms such as \( (a-x^n)^m \), where direct integration can be cumbersome.
Definite integrals
Definite integrals are used to calculate the area under a curve for a given interval. They have the form \[\int_{a}^{b} f(x) \, dx\]and are evaluated between two specific bounds, \( a \) and \( b \). Here鈥檚 how definite integrals are computed:
  • First, find the antiderivative (or indefinite integral) of the function \( f(x) \).
  • Next, evaluate the antiderivative at the upper limit \( b \) and then at the lower limit \( a \).
  • Finally, subtract the value of the antiderivative at \( a \) from its value at \( b \).
The result gives the net area between the curve of the function and the x-axis, from \( a \) to \( b \). Definite integrals can tell us more than just areas鈥攕uch as the displacement in physics or total accumulated value in economics鈥攄epending on the context of the problem.
Indefinite integrals
Indefinite integrals are used to find the antiderivative of a function. Unlike definite integrals, they don鈥檛 have upper or lower limits. Their general form is\[\int f(x) \, dx = F(x) + C\]Here, \( F(x) \) is the antiderivative, and \( C \) represents the constant of integration. This constant is important because integration is essentially the reverse of differentiation. The process of finding indefinite integrals involves:
  • Identifying forms that are straightforward or require integration techniques like substitution, integration by parts, or partial fraction decomposition.
  • Computing the antiderivative \( F(x) \) by using these methods and rules such as the power rule, summation rule, etc.
  • Adding the constant \( C \) because differentiation of constants is always zero, acknowledging this family of functions that differ only by a constant.
Indefinite integrals are incredibly useful for solving differential equations, modeling accumulation functions, and understanding the behavior of functions.

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Most popular questions from this chapter

Make the \(u\) -substitution and evaluate the resulting definite integral. $$ \begin{aligned} &\int_{0}^{+\infty} \frac{e^{-x}}{\sqrt{1-e^{-x}}} d x ; u=1-e^{-x} \\ &{[\text { Note: } u \rightarrow 1 \text { as } x \rightarrow+\infty .]} \end{aligned} $$

Make the \(u\) -substitution and evaluate the resulting definite integral. $$ \int_{0}^{+\infty} \frac{e^{-x}}{\sqrt{1-e^{-2 x}}} d x ; u=e^{-x} $$

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

Engineers want to construct a straight and level road \(600 \mathrm{ft}\) long and \(75 \mathrm{ft}\) wide by making a vertical cut through an intervening hill (see the accompanying figure). Heights of the hill above the centerline of the proposed road, as obtained at various points from a contour map of the region, are shown in the accompanying figure. To estimate the construction costs, the engineers need to know the volume of earth that must be removed. Approximate this volume, rounded to the nearest cubic foot. [Hint: First set up an integral for the cross-sectional area of the cut along the centerline of the road, then assume that the height of the hill does not vary between the centerline and edges of the road.] $$ \begin{array}{cc} \hline \begin{array}{l} \text { HORIZONTAL } \\ \text { DISTANCE } x \text { (ft) } \end{array} & \begin{array}{c} \text { HEIGHT } \\ h(\mathrm{ft}) \end{array} \\ \hline 0 & 0 \\ 100 & 7 \\ 200 & 16 \\ 300 & 24 \\ 400 & 25 \\ 500 & 16 \\ 600 & 0 \end{array} $$

Some integrals that can be evaluated by hand cannot be evaluated by all computer algebra systems. Evaluate the integral by hand, and determine if it can be evaluated on your CAS. $$ \int \frac{1}{x^{10}+x} d x $$
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