Integration Strategies

U Substitution

1. Identify $u$ (might need to try multiple times until you find one that cancels)
2. Find $du$
3. Substitute in to original (+ change bounds if discrete)
4. Integrate in terms of $u$
5. Plug back in for $u$ (or just plug in the bounds if discrete)

Long Division

For a rational function where the degree of the numerator is greater or equal to the degree of the denominator

Inverse Trig

• Can manipulate to get an integral in terms of the arctan or arcsin rule
• You can complete the square to get an integral in the form of arctan rule

Partial Fraction Decomposition

• For rational functions where there is a bigger degree on the bottom
• Use Partial Fraction Decomposition and then integrate

Integration By Parts

Reverses the product rule

Normal Method

$\int u{v}^{\prime }dx=uv-\int u^{\prime }vdx$

• Strategies for picking u and v
  • let dv be the more complicated part of the integral that can be easily integrated
  • let u be the part of the integrand whose derivative is a simples function than u (or at least not more complicated
  • choose u to be LIATE: log, inverse trig, algebraic, trig, exponential
  • if one doesn’t work for u, chose the other

Tabular Method

• For repeated integration by parts
• Steps
  1. Create a table with 3 columns
     • S (alternating sign)
     • D (derivative of u)
     • I (antiderivative of dv)
  2. Fill out columns until D = 0
  3. Your solution is $\sum _{n=0}^{rows-1}{S}_n\cdot D_n\cdot I_{n+1}$
     • Diagonal lines
• If column D is not going to get to 0
  1. look for the appearance of the original integral
  2. write out what you have so far so the original integral is the $-\int u\prime vdx$ 3. add that to the other side and then divide by 2

Trig Substitution

A way to eliminate roots in integrands

Given	Let	Theta In	Result
$\sqrt{a^2-b^2{x}^2}$	$x=a\sin \theta$	$[-\frac{\pi }{2},\frac{\pi }{2}]$	$\frac{a}{b}\cos \theta$
$\sqrt{a^2+b^2{x}^2}$	$x=a\tan \theta$	$[-\frac{\pi }{2},\frac{\pi }{2}]$	$\frac{a}{b}\sec \theta$
$\sqrt{x^2-b^2{a}^2}$	$x=a\sec \theta$	$[0,\frac{\pi }{2})\cup [\pi ,\frac{3\pi }{2})$	$\frac{a}{b}\tan \theta$

Higher Power Trig Handy Things

• Odd
  1. Split Up
  2. Pythagorean Identity
  3. U-Sub and cancel
• Even
  • ${\cos }^2\theta =\frac{1}{2}(1+\cos (2\theta ))$
  • ${\sin }^2\theta =\frac{1}{2}(1-\cos (2\theta ))$