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Chapter 8: Problem 40

Use integration tables to find the integral. $$ \int \sqrt{\frac{3-x}{3+x}} d x $$

Short Answer

Expert verified
The result of the integral is \(2\sqrt{-x^2} + C\) .

Step by step solution

01

Perform a variable substitution

Introduce a new variable via substitution: Let \(u = 3 + x\). The derivative of \(u\) with respect to \(x\) is \(du/dx = 1\), and \(dx = du\). So the integral can be rewritten as \(\int \sqrt{\frac{3-u}{u}} du\).
02

Rewrite the integral

The integral in \(u\) takes the form of a standard integral that can be found in most integral tables. Rewrite it as: \(\int u^{-0.5}(3-u)^{0.5} du\). This is a form that can be identified as a standard integral form.
03

Find the integral using a standard integral table

The integral can now be resolved using common integration tables, or using online calculus tools. Solving this results in \(2\sqrt{u(3-u)}\). After the integration, don't forget to substitute u back in terms of x.
04

Substituting back to x

Substitute u back in terms of x, we have \(2\sqrt{(3+x)(3-(3+x))}\), which simplifies to \(2\sqrt{-x^2}\). Remember to include +C in the result since this is an indefinite integral.

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Key Concepts

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

Integration Tables
Integration tables are helpful tools for solving integrals when you encounter forms that fit standard patterns. These tables provide ready-to-use antiderivatives of various functions, which saves time and effort in the integration process.
These tables usually list indefinite integrals for:
  • Polynomial functions
  • Trigonometric functions
  • Exponential and logarithmic functions
  • Rational functions
By matching the function you need to integrate with a similar form in the table, you can directly write down its integral.
In our given exercise, the integral \[\int \sqrt{\frac{3-x}{3+x}} dx\]can be rewritten to match a standard form using integration tables.
Variable Substitution
Variable substitution is a powerful technique used in integration to make the integral easier. It's like changing clothes on a mathematical problem to make it fit a form you can solve.
The goal here is to simplify the integrand (the function being integrated) by introducing a new variable. In our exercise, setting
  • \(u = 3 + x\)
helps reformat the integral to something you can manage more easily.
By substituting, you also change the differential. Here, \(du/dx = 1\), so \(dx = du\). Your integral changes,\[\int \sqrt{\frac{3-u}{u}} du\], allowing you to solve it using tools like an integration table.
Integral Calculus
Integral calculus focuses on finding the antiderivative or the original function, given its derivative. It wants you to know, "From what did this come?" In your exercise, you started with a complex expression needing integration.
This branch of calculus serves two main purposes:
  • Determining the area under a curve, which is exactly what an integral represents.
  • Recovering the original function from its derivative, informing you of initial conditions.
In indefinite integrals like\[\int \sqrt{\frac{3-x}{3+x}} dx\],you're looking for a function whose derivative gives you back the initial expression. After substituting and integrating, you must remember to revert your variable back from \(u\) to \(x\) and add the integration constant \(+C\) for completeness.

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Most popular questions from this chapter

Laplace Transforms Let \(f(t)\) be a function defined for all positive values of \(t\). The Laplace Transform of \(f(t)\) is defined by$$F(s)=\int_{0}^{\infty} e^{-s t} f(t) d t$$if the improper integral exists. Laplace Transforms are used to solve differential equations, find the Laplace Transform of the function. $$ f(t)=t $$

$$ \text { Evaluate } \int_{0}^{\pi / 2} \frac{d x}{1+(\tan x)^{\sqrt{2}}} $$

True or False? , determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false. If \(f\) is continuous on \([0, \infty)\) and \(\lim _{x \rightarrow \infty} f(x)=0\), then \(\int_{0}^{\infty} f(x) d x\) converges.

(a) Use a graphing utility to graph the function \(y=e^{-x^{2}}\). (b) Show that \(\int_{0}^{\infty} e^{-x^{2}} d x=\int_{0}^{1} \sqrt{-\ln y} d y\).

Consider the region satisfying the inequalities. (a) Find the area of the region. (b) Find the volume of the solid generated by revolving the region about the \(x\) -axis. (c) Find the volume of the solid generated by revolving the region about the \(y\) -axis. $$ y \leq \frac{1}{x^{2}}, y \geq 0, x \geq 1 $$
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