Essential Concepts

Integration by Parts

• The integration-by-parts formula allows the exchange of one integral for another, possibly easier, integral.
• Integration by parts applies to both definite and indefinite integrals.

Trigonometric Integrals

• Integrals of trigonometric functions can be evaluated by the use of various strategies. These strategies include:
  1. Applying trigonometric identities to rewrite the integral so that it may be evaluated by u-substitution
  2. Using integration by parts
  3. Applying trigonometric identities to rewrite products of sines and cosines with different arguments as the sum of individual sine and cosine functions
  4. Applying reduction formulas
• For integrals involving [latex]\sqrt{{a}^{2}-{x}^{2}}[/latex], use the substitution [latex]x=a\sin\theta[/latex] and [latex]dx=a\cos\theta d\theta[/latex].
• For integrals involving [latex]\sqrt{{a}^{2}+{x}^{2}}[/latex], use the substitution [latex]x=a\tan\theta[/latex] and [latex]dx=a{\sec}^{2}\theta d\theta[/latex].
• For integrals involving [latex]\sqrt{{x}^{2}-{a}^{2}}[/latex], substitute [latex]x=a\sec\theta[/latex] and [latex]dx=a\sec\theta \tan\theta d\theta[/latex].

Partial Fractions

• Partial fraction decomposition is a technique used to break down a rational function into a sum of simple rational functions that can be integrated using previously learned techniques.
• When applying partial fraction decomposition, we must make sure that the degree of the numerator is less than the degree of the denominator. If not, we need to perform long division before attempting partial fraction decomposition.
• The form the decomposition takes depends on the type of factors in the denominator. The types of factors include nonrepeated linear factors, repeated linear factors, nonrepeated irreducible quadratic factors, and repeated irreducible quadratic factors.

Other Strategies for Integration

• An integration table may be used to evaluate indefinite integrals.
• A CAS (or computer algebra system) may be used to evaluate indefinite integrals.
• It may require some effort to reconcile equivalent solutions obtained using different methods.

Key Equations

• Integration by parts formula
  
  [latex]\displaystyle\int udv=uv-\displaystyle\int vdu[/latex]
• Integration by parts for definite integrals
  
  [latex]{\displaystyle\int }_{a}^{b}udv={uv|}_{a}^{b}-{\displaystyle\int }_{a}^{b}vdu[/latex]

To integrate products involving [latex]\sin\left(ax\right)[/latex], [latex]\sin\left(bx\right)[/latex], [latex]\cos\left(ax\right)[/latex], and [latex]\cos\left(bx\right)[/latex], use the substitutions.

• Sine Products
  
  [latex]\sin\left(ax\right)\sin\left(bx\right)=\frac{1}{2}\cos\left(\left(a-b\right)x\right)-\frac{1}{2}\cos\left(\left(a+b\right)x\right)[/latex]
• Sine and Cosine Products
  
  [latex]\sin\left(ax\right)\cos\left(bx\right)=\frac{1}{2}\sin\left(\left(a-b\right)x\right)+\frac{1}{2}\sin\left(\left(a+b\right)x\right)[/latex]
• Cosine Products
  
  [latex]\cos\left(ax\right)\cos\left(bx\right)=\frac{1}{2}\cos\left(\left(a-b\right)x\right)+\frac{1}{2}\cos\left(\left(a+b\right)x\right)[/latex]
• Power Reduction Formula
  
  [latex]{\displaystyle\int}{\text{sec}}^{n}xdx=\frac{1}{n - 1}{\text{sec}}^{n - 1}x+\frac{n - 2}{n - 1}{\displaystyle\int}{\text{sec}}^{n - 2}xdx[/latex]
• Power Reduction Formula
  
  [latex]{\displaystyle\int}{\tan}^{n}xdx=\frac{1}{n - 1}{\tan}^{n - 1}x-{\displaystyle\int}{\tan}^{n - 2}xdx[/latex]

Glossary

computer algebra system (CAS)

technology used to perform many mathematical tasks, including integration

integration by parts
a technique of integration that allows the exchange of one integral for another using the formula [latex]{\displaystyle\int}udv=uv-{\displaystyle\int}vdu[/latex]

integration table

a table that lists integration formulas

partial fraction decomposition

a technique used to break down a rational function into the sum of simple rational functions

power reduction formula
a rule that allows an integral of a power of a trigonometric function to be exchanged for an integral involving a lower power

trigonometric integral

an integral involving powers and products of trigonometric functions

trigonometric substitution

an integration technique that converts an algebraic integral containing expressions of the form [latex]\sqrt{{a}^{2}-{x}^{2}}[/latex], [latex]\sqrt{{a}^{2}+{x}^{2}}[/latex], or [latex]\sqrt{{x}^{2}-{a}^{2}}[/latex] into a trigonometric integral