周次: 7 日期: April 6, 2022 节次: 1

In this lecture, we explore more examples of integration. Trigonometric substitutions and integration by looking up the integration tables are covered. Note that most of these calculations can be done by hands.

§ 8.5 Three Basic Substitutions

The most common substitutions are $x=a\tan\theta$, $a=\sin\theta$, and $x=a\sec\theta$. They can be useful for dealing with the integrals involving $\sqrt{a^2-x^2}$, $\sqrt{a^2+x^2}$ and $\sqrt{x^2-a^2}$.

  • With integrals involving $\sqrt{a^2-x^2}$, we often use the substitution $x=a\sin(\theta)$ (or $x=a\cos(\theta)$)
  • With integrals involving $\sqrt{a^2+x^2}$, we often use the substitution $x=a\tan(\theta)$
  • With integrals involving $\sqrt{x^2-a^2}$, we often use the substitution $x=a \sec(\theta)$ (For the integrals involving $\sqrt{x^2-a^2}$ and $\sqrt{x^2+a^2}$ we can also apply the known result $\int\frac{dx}{\sqrt{x^2\pm a^2}}=\ln |x+\sqrt{x^2\pm a^2}| + C$)

Remark As doing these substitutions, we need them to be reversible, so that we can change back to the original variable afterward.

For reversibility,

  • $x=a\tan\theta$ requires $\theta = \arctan \left(x/a\right)$, with $-\frac{\pi}{2}<\theta<\frac{\pi}{2}$,
  • $x = a \sin \theta$ requires $\theta = \arcsin(x/a)$, with $-\frac{\pi}{2}\le \theta \le \frac{\pi}{2}$,
  • $x=a\sec \theta$ requires $\theta = \mathrm{arcsec}(x/a)$, with $\begin{cases}0\le \theta < \frac\pi2 & \text{if }x/a\ge 1,\ \frac\pi2 < \theta \le \pi & \text{if } x/a \le -1.\end{cases}$

Example 1. Using the Substitution $x=a\tan\theta$

Evaluate

$$ \int \dfrac{dx}{\sqrt{4+x^2}} $$

Solution. $\ln |\sqrt{4+x^2}+x|+ C.$

Example 2. Using the Substitution $x=a\sin\theta$

Evaluate

$$ \int \dfrac{x^2 dx}{\sqrt{9-x^2}} $$

Solution. $\frac{9}{2}\arcsin\frac{x}{3}-\frac{x}{2}\sqrt{9-x^2}+C$

Example 3. Using the Substitution $x=a\sec\theta$

Evaluate

$$ \int \dfrac{dx}{\sqrt{25x^2-4}}, \quad x> \frac{5}{2}. $$

Solution. $\frac15 \ln \left|\frac{5x}{2}+\frac{\sqrt{25x^2-4}}{2}\right| + C.$

Example 4. Finding the Volume of a Solid of Revolution

Find the volume of the solid generated by revolving about the $x$-axis the region bounded by the curve $y=\dfrac{4}{x^2+4}$, the $x$-axis, and the lines $x=0$ and $x=2.$

Solution. $\pi\left[\frac\pi4 + \frac12\right]$.

Example 5. Finding the Area of an Ellipse

Find the area enclosed by the ellipse

$$ \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1. $$

Solution. $\pi a b$.

§ 8.6 Integral Tables and Computer Algebra Systems

Integral Tables

A Brief Table of Integrals is provided at the back of the book. The integration formulas are stated in terms of constants $a$, $b$, $c$, $m$, $n$ and so on. The limitations on their values are stated with the formulas. The formulas also assume that the constants do not take on values that require dividing by $0$ or taking even roots of negative numbers.

Note that the following examples can also be evaluated using algebraic manipulation, substitution or integration by parts.

Example 1. Find

$$ \int \dfrac{x}{2x+5}dx $$

Example 2. Find

$$ \int \frac{dx}{x\sqrt{2x+4}} $$

Example 3. Find

$$ \int \dfrac{dx}{x\sqrt{2x-4}} $$

Example 4. Find

$$ \int\frac{dx}{x^2\sqrt{2x-4}} $$

Example 5. Find

$$ \int x \arcsin x dx $$

Reduction Formulas

Some integrals can be reduced by integration by parts, for example

Example 6. Reduce the Formula

Find

$$ \int \tan^n x dx, $$

and evaluate the integral $\int \tan^5 dx.$

Solution. $\frac1{n-1}\tan^{n-1}(x) - \int \tan^{n-2}xdx$, and $\frac{1}4\tan^4 x -\frac12 \tan^2 x -\ln|\cos x |+ C$Exa

Example 7. Deriving a Reduction Formula

Show that for any positive integer $n$

$$ \int(\ln x)^ndx = x\ln(x)^n-n\int (\ln x)^{n-1} dx. $$

Example 8. Find

$$ \int \sin^2 x \cos^3 xdx. $$

Note that this integrand has an odd power in $\cos$ part, therefore can be evaluated by substitution $u=\sin(x)$. The integral can also be evaluated by looking up the integration table.

Non-elementary Integrals

The integrals of functions that do not have elementary antiderivatives are called non-elementary integrals. For example,

$$ \int e^{-t^2}dt,\quad \int \sin x^2 dx, \quad \int \sqrt{1+x^4}dx, \int \frac{e^x}{x}dx,\quad \int \frac{1}{\ln x}dx, \quad \int \frac{\sin x}{x}dx, \ \int\sqrt{1-k^2\sin^2 x}dx, \quad 0<k<1. $$

Integration with a CAS (Computer Algebraic System)

For example, the integrations can be evaluated by using Mathematica, Maple , or Python (SymPy). Here we illustrate the use of Mathematica and Python.

Example. Integrating via Mathematica.

$$ \int x^2 \sqrt{a^2+x^2}dx. $$

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In Python, one has to import SymPy first by using

from sympy import *

To perform any integration, first one needs to declare the variable.

u = Symbol('u')

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Note that sometimes CASs may give totally different looking results.

Example 10. Integrate

$$ \int \sin^2 x \cos^3 x dx $$

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Example 11. Integrate

$$ \int \arccos(ax)^2 dx. $$

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