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Chapter 7: Problem 63

(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). $$ \int \frac{x^{3}}{\sqrt{1+x^{2}}} d x $$

Short Answer

Expert verified
The integral evaluates to \( \frac{(1+x^2)^2}{4} - \frac{1+x^2}{2} + C \).

Step by step solution

01

Choose a Suitable Substitution

Let's use the substitution method to solve the integral. We look for a substitution that can simplify the integrand. Here, we can use the substitution \( u = \sqrt{1+x^2} \). This choice is suitable because the derivative of \( x^2 \), i.e., \( 2x \), appears as a factor within the integral.
02

Compute the Derivative and Differential

First, express \( u \) in terms of \( x \): let \( u = (1 + x^2)^{1/2} = \sqrt{1+x^2} \). Now differentiate both sides with respect to \( x \): \( \frac{du}{dx} = \frac{x}{\sqrt{1+x^2}} \). Thus, \( du = \frac{x}{\sqrt{1+x^2}} dx \), or equivalently, \( x \cdot dx = \sqrt{1+x^2} \cdot du \). This will help us in substituting the integrand correctly.
03

Substitute and Simplify the Integral

Using our substitution, the integral becomes: \[ \int \frac{x^3}{\sqrt{1+x^2}} \, dx = \int x^2 \cdot x \cdot dx = \int (u^2 - 1) u \, du. \] This simplification is obtained by recognizing that \( x^2 = u^2 - 1 \) from the relation \( u^2 = 1 + x^2\). Now substitute these into the integral.
04

Evaluate the Integral

The integral can now be rewritten as \( \int (u^2 - 1) u \, du = \int (u^3 - u) \, du \). We will integrate this term-by-term resulting in: \[ \frac{u^4}{4} - \frac{u^2}{2} + C, \] where \( C \) is the constant of integration.
05

Substitute Back to Original Variables

Substitute back \( u = \sqrt{1+x^2} \) into the antiderivative, we have: \[ \frac{(\sqrt{1+x^2})^4}{4} - \frac{(\sqrt{1+x^2})^2}{2} + C = \frac{(1+x^2)^2}{4} - \frac{1+x^2}{2} + C. \] Simplify if needed.

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Key Concepts

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

Integration Techniques
Calculus is a field of mathematics that focuses on understanding change and motion through two main concepts: differentiation and integration. Integration techniques are crucial for tackling complex integrals that can't be solved at a glance. Among the various methods, substitutions, especially using the chain rule, can be very useful. They allow the transformation of challenging integrals into simpler forms.

When solving integrals, identifying a part of the integrand that can act as a substitute (often denoted as "u") simplifies the integration. The process involves:
  • Choosing a suitable substitution that matches closely with part of the integrand.
  • Computing the derivative of the substitution, often necessary to convert dx to du.
  • Reconfiguring the integral in terms of "u," and solving it before returning to the variable "x."
This method requires practice to master, as the choice of substitution can greatly affect the ease of solving the integral.
u-Substitution
u-Substitution is a specific integration technique that simplifies the process by substituting a part of the integrand with a single variable, usually denoted as "u." This acts as a temporary replacement to simplify differentiation and integration calculations.

The process involves three essential steps:
  • Identifying an inner function within the integrand to act as "u."
  • Computing its derivative, or the derivative of a related term, allowing conversion of all parts of the integrand to involve "u."
  • Rewriting the integral in terms of "u" and integrating with respect to "u."
After integration, the original variable is reintroduced by substituting "u" back into the expression. The challenge lies in recognizing effective substitutions, often guided by looking for components in the integrand whose derivatives are also present or can easily form part of the "du." In the example problem, the substitution of \( u = \sqrt{1+x^2} \) allowed a simplification due to the presence of \( x \), which was compatible with the derived differential expression.
Definite Integrals
Definite integrals offer a way to calculate the net area under a curve between two points on a graph. While evaluating definite integrals, we typically replace indefinite integral constants with specific limits of integration. This turns the antiderivative into a function with clear boundaries, allowing computation of exact numerical values.

To solve a definite integral:
  • Compute the indefinite integral first, either by simple integration or using techniques like substitution.
  • Apply the Fundamental Theorem of Calculus, evaluating the antiderivative at the upper limit and subtracting the evaluation at the lower limit.
In the solved example, the integral was initially indefinite. However, if limits were provided, re-substituted terms would be evaluated at these bounds. Proper application of definite integrals is essential in many scientific calculations, including computing areas, volumes, and other constants in physical and engineering contexts.

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Most popular questions from this chapter

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). $$ \int x \sqrt{x-2} d x $$

The exact value of the given integral is \(\pi\) (verify). Approximate the integral using (a) the midpoint approximation \(M_{10},(\mathrm{~b})\) the trapezoidal approximation \(T_{10}\), and (c) Simpson's rule approximation \(S_{20}\) using Formula (7). Approximate the absolute error and express your answers to at least four decimal places. $$ \int_{0}^{2} \frac{8}{x^{2}+4} d x $$

A transform is a formula that converts or "transforms" one function into another. Transforms are used in applications to convert a difficult problem into an easier problem whose solution can then be used to solve the original difficult problem. The Laplace transform of a function \(f(t)\), which plays an important role in the study of differential equations, is denoted by \(\mathscr{L}\\{f(t)\\}\) and is defined by $$\mathscr{L}\\{f(t)\\}=\int_{0}^{+\infty} e^{-s t} f(t) d t$$ In this formula \(s\) is treated as a constant in the integration process; thus, the Laplace transform has the effect of transforming \(f(t)\) into a function of \(s .\) Use this formula in these exercises. In each part, find the Laplace transform. (a) \(f(t)=t, s>0\) (b) \(f(t)=t^{2}, s>0\) (c) \(f(t)=\left\\{\begin{array}{ll}0, & t<3 \\ 1, & t \geq 3\end{array}, s>0\right.\)

Evaluate the integral. $$ \int_{0}^{\pi / 2} \tan ^{5} \frac{x}{2} d x $$
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