Calculus/Further Methods of Integration/Contents

Further Methods of Integration

In this chapter, you will study several integration techniques which will greatly expand the set of integrals to which you can find a closed-form indefinite integral. Probably the most difficult step in integrating a differential function is recognizing the proper formula to use. This will come with practice.

1. Basic Integration Rules
2. Integration by Tables and Reducation Formulae
3. Partial Fractions
4. Trigonometric Integrals
5. Trigonometric Substitution
6. Improper Integrals

Review

Basic Integration Rules

$\int 0du=C$

$\int kudu=k\times \int udu+C$

$\int (u+v)du=\int udu+\int vdu+C$

Trigonometric Integrals

$\int \sin xdx=-\cos x+C$

$\int \cos xdx=\sin x+C$

$\int \sec xdx=\ln |\sec x+\tan x|+C$

$\int \csc xdx=-\ln |\csc x+\cot x|+C$

$\int \tan xdx=-\ln |\cos x|+C$

$\int \cot xdx=\ln |\sin x|+C$

$\int \sec x\tan xdx=\sec x+C$

$\int \csc x\cot xdx=-\csc x+C$

$\int {\sec }^{2}xdx=\tan x+C$

$\int {\csc }^{2}xdx=-\cot x+C$

Partial Integration

For two functions u and dv of a variable x,

$\int udv=uv-\int vdu$

where u is chosen by precedence according to LIPET:

• Logarithmic
• Inverse Trigonometric
• Polynomial
• Exponential
• Trigonometric

Improper Integrals

For any function f of variable x, continuous on the given infinite domain:

$\underset{a}{\overset{\mathrm{\infty }}{\int }}f(x)dx$=$\underset{b\to \mathrm{\infty }}{\lim }\underset{a}{\overset{b}{\int }}f(x)dx$

$\underset{-\mathrm{\infty }}{\overset{b}{\int }}f(x)dx$=$\underset{a\to -\mathrm{\infty }}{\lim }\underset{a}{\overset{b}{\int }}f(x)dx$

$\underset{-\mathrm{\infty }}{\overset{\mathrm{\infty }}{\int }}f(x)dx$=$\underset{-\mathrm{\infty }}{\overset{c}{\int }}f(x)dx+\underset{c}{\overset{\mathrm{\infty }}{\int }}f(x)dx$

For any function f of variable x continuous on the given interval, but with an infinite discontinuity at (1) a, (2) b, or some (3) c in [a,b]:

$\underset{a}{\overset{b}{\int }}f(x)dx$=$\underset{c\to b^{-}}{\lim }\underset{a}{\overset{c}{\int }}f(x)dx$ (1)

$\underset{a}{\overset{b}{\int }}f(x)dx$=$\underset{c\to a^{+}}{\lim }\underset{c}{\overset{b}{\int }}f(x)dx$ (2)

$\underset{a}{\overset{b}{\int }}f(x)dx$=$\underset{a}{\overset{c}{\int }}f(x)dx+\underset{c}{\overset{b}{\int }}f(x)dx$ (3) Template:Calculus:TOC