Objectives
- Apply the various techniques of integration to different integrals
Section 7.5 Strategy for Integration
Subsection 7.5.1 Before Class
https://mymedia.ou.edu/media/7.5-1/1_ahe8kp1mSubsubsection 7.5.1.1 Integrals We Know
Collected below are all of the integrals that we have learned:
| Function | Antiderivative | Function | Antiderivative |
| \(x^n\) | \(\dfrac{1}{n+1} x^{n+1} + C\) for \(x\neq -1\) | \(\dfrac{1}{x}\) | \(\ln |x| + C\) |
| \(e^x\) | \(e^x + C\) | \(b^x\) | \(\dfrac{1}{\ln b} b^x + C\) |
| \(\sin x\) | \(-\cos x + C\) | \(\cos x\) | \(\sin x + C\) |
| \(\sec^2 x\) | \(\tan x + C\) | \(\csc^2 x\) | \(-\cot x + C\) |
| \(\sec x \tan x\) | \(\sec x + C\) | \(\csc x \cot x\) | \(-\csc x\) |
| \(\sec x\) | \(\ln |\sec x + \tan x| + C\) | \(\csc x\) | \(\ln |\csc x - \cot x| + C\) |
| \(\tan x\) | \(\ln |\sec x| +C \) | \(\cot x\) | \(\ln |\sin x | + C\) |
| \(\dfrac{1}{x^2 + a^2}\) | \(\dfrac{1}{a}\inv{\tan}\lrpar{\dfrac{x}{a}} + C\) | \(\dfrac{1}{\sqrt{a^2-x^2}}\) | \(\inv{\sin}\lrpar{\dfrac{x}{a}}\) |
Subsubsection 7.5.1.2 Strategies
When faced with an integral, it can be difficult to determine which method is the best to use. This flowchart may provide some help:
- Simplify the integrand: try algebraic simplifications and/or use trig identities
- Try u-substitution: look for "obvious" choices
- Classify the integral: 1) Trig functions, 2) Rational functions, 3) Integration by Parts, 4) A radical
-
Try again:
- Be creative with u-substitution
- Integration by parts
- Less obvious algebraic manipulations, like multiplying by a form of 1
- Try to relate the integral to one you’ve done before
- Try multiple methods
Example 7.5.1.
Find \(\ds \int \dfrac{1}{1+\cos x}\, dx \)
Solution.
\(\ds \int \dfrac{1}{1+\cos x}\, dx =-\cot x + \csc x + C\)
Example 7.5.2.
Compute \(\ds \int e^{\sqrt{x}}\, dx\)
Solution.
\(\ds \int e^{\sqrt{x}}\, dx = 2\sqrt{x}e^{\sqrt{x}} - 2e^{\sqrt{x}} + C\)
Example 7.5.3.
Compute \(\ds \int \sqrt{\dfrac{1+x}{1-x}}\, dx\)
Solution.
\(\ds \int \sqrt{\dfrac{1+x}{1-x}}\, dx = \inv{\sin}(x) + \sqrt{1-x^2} + C\)
Subsection 7.5.2 Pre-Class Activities
Example 7.5.4.
Write any questions you have from the videos in this space.
Solution.
Answers vary
Example 7.5.5.
Analyze the following integrals using the flow chart from above, determine what you think the best approach is, and briefly write why. Do not compute these integrals! This exercise is here to help you practice analyzing situations. We’ll do these problems in class.
- \(\displaystyle \ds \int \dfrac{\cos x}{1-\sin x}\, dx\)
- \(\displaystyle \ds \int \dfrac{\sin^3 x}{\cos x}\, dx\)
- \(\displaystyle \ds \int \dfrac{t}{t^4 + 2}\, dt\)
- \(\displaystyle \ds \int_2^4 \dfrac{x+2}{x^2 + 3x - 4}\, dx\)
- \(\displaystyle \ds \int x\sec x\tan x\, dx\)
Solution.
Answers vary for all integrals; the important part here is to think about techniques
Subsection 7.5.3 In Class
Subsubsection 7.5.3.1 Examples
Example 7.5.6.
\(\ds \int \dfrac{\cos x}{1-\sin x}\, dx\)
Solution.
\(\ds \int \dfrac{\cos x}{1-\sin x}\, dx = \ln |\sec x+ \tan x| + \ln |\sec x| + C\)
Example 7.5.7.
\(\ds \int \dfrac{\sin^3 x}{\cos x}\, dx\)
Solution.
\(\ds \int \dfrac{\sin^3 x}{\cos x}\, dx = \ln |\sec x| + \dfrac{1}{4}\cos 2x + C\)
Example 7.5.8.
\(\ds \int \dfrac{t}{t^4 + 2}\, dt\)
Solution.
\(\ds \int \dfrac{t}{t^4 + 2}\, dt = \dfrac{\sqrt{2}}{4}\inv{\tan}\lrpar{\dfrac{t^2}{\sqrt{2}}}+ C\)
Example 7.5.9.
\(\ds \int_2^4 \dfrac{x+2}{x^2 + 3x - 4}\, dx\)
Solution.
\(\ds \int_2^4 \dfrac{x+2}{x^2 + 3x - 4}\, dx = \dfrac{2}{5}\ln 8 + \dfrac{3}{5}\ln 3 - \dfrac{2}{5}\ln 6\)
Example 7.5.10.
\(\ds \int x\sec x\tan x\, dx\)
Solution.
\(\ds \int x\sec x\tan x\, dx = x\sec x - \ln \sec x + \tan x| + C\)
Example 7.5.11.
\(\ds \int_0^1 \dfrac{x}{(2x+1)^3}\, dx\)
Solution.
\(\ds \int_0^1 \dfrac{x}{(2x+1)^3}\, dx = -\dfrac{4x+1}{8(2x+1)^2} + C\)
Example 7.5.12.
\(\ds \int \ln(1+x^2)\, dx\)
Solution.
\(\ds \int \ln(1+x^2)\, dx = x\ln (x^2 + 1) - 2x + 2\arctan x + C\)
Example 7.5.13.
\(\ds \int \dfrac{\ln x}{x\sqrt{1+(\ln x)^2}}\, dx\)
Solution.
\(\ds \int \dfrac{\ln x}{x\sqrt{1+(\ln x)^2}}\, dx = \sqrt{(\ln x)^2 + 1} + C\)
Example 7.5.14.
\(\ds \int (1+\tan x)^2\sec x\, dx\)
Solution.
\(\ds \int (1+\tan x)^2\sec x\, dx = \dfrac{1}{2}\ln |\sec x + \tan x| + \dfrac{1}{2}\sec x\tan x + 2 + C\)
Example 7.5.15.
\(\ds \int \sqrt{3-2x-x^2}\, dx\)
Solution.
\(\ds \int \sqrt{3-2x-x^2}\, dx = 2\inv{\sin}\lrpar{\dfrac{x+1}{2}} + \dfrac{(x+1)}{2}\sqrt{4-(x+1)^2} + C\)
Example 7.5.16.
\(\ds \int \dfrac{\inv{\tan}(x)}{x^2}\, dx\)
Solution.
\(\ds \int \dfrac{\inv{\tan}(x)}{x^2}\, dx = -\dfrac{\inv{\tan}(x)}{x} - \dfrac{1}{2}\ln\lrpar{\dfrac{x^2+1}{x^2}} + C\)
Example 7.5.17.
\(\ds \int_0^{\pi/4} \tan^3\theta\sec^2\theta\, d\theta\)
Solution.
\(\ds \int_0^{\pi/4} \tan^3\theta\sec^2\theta\, d\theta = \dfrac{1}{4}\)
Example 7.5.18.
\(\ds \int \dfrac{x + \arcsin x}{\sqrt{1-x^2}}\, dx\)
Solution.
\(\ds \int \dfrac{x + \arcsin x}{\sqrt{1-x^2}}\, dx = \dfrac{1}{2}\arcsin^2(x) - \sqrt{1-x^2} + C\)
Example 7.5.19.
\(\ds \int \dfrac{4^x + 10^x}{2^x}\, dx\)
Solution.
\(\ds \int \dfrac{4^x + 10^x}{2^x}\, dx = \dfrac{1}{\ln 2}2^x + \dfrac{1}{\ln 5}5^x + C\)
Example 7.5.20.
\(\ds \int e^2\, dx\)
Solution.
\(\ds \int e^2\, dx = e^2x + C\)
Subsection 7.5.4 After Class Activities
Example 7.5.21.
Look back at the examples we did in class. Make sure that you can follow the thought process that led us to use that particular integration technique.
Solution.
Answers vary
Example 7.5.22.
Refer back to the strategy list. Try to give an example of an integral which fits with each step.
Solution.
Answers vary
Example 7.5.23.
Which technique do you feel like you need the most practice with? Why?
Solution.
Answers vary
Example 7.5.24.
There are many practice problems available in the book, on page 548. Work through as many as you can; the more practice you get, the more confident and capable you’ll be!
