Strategies For Integration

Strategies For Integration

1. 1. Simplify the integrand, if possible.

   • very easy with a little simplification or manipulation.
     • Algebraic Manipulation
       • Examples
         • ∫√ x(1 + √ x)dx = ∫( √ x + x)dx
     • Trigonometric Identities
       • Example
         • ∫ tan θ/sec^2 θ dθ = ∫ (sin θ/cos θ) cos^2 θdθ = ∫ sin θ cos θdθ = 1/2 ∫sin 2θdθ

2. 2. See if a “simple” substitution will work.

   • Example
     • ∫x/(x^2 − 1) dx
       • use u = x^2 − 1 and 1/2 du = xdx

3. 3. Identify the type of integral.

   • Trigonometric functions/Inverse Trigonometric Function/Natural Logarithmic/Exponential Function/Hyperbolic function/Inverse Hyperbolic Function
     • Use substitutions method
   • Radicals
     • Form √ ±x^2 ± a^2, Use trigonometric substitution method
   • Integration by parts
     • Use integration by parts when the integrand is a product of two functions, such as a power of x and a trigonometric, exponential or logarithmic function.
       • I L A T E RULE
   • Rational functions.
     • Use the method of partial fractions
   • Trigonometric Integrals
     • Use Half-Angle formula
     • Use Reduction Formula
   • Improper Integral
     • Use limit technique
   • Products of Power of Sines and Cosines
     • Use trigonometric Identites
   • Products of Sines and Cosines
     • Use trigonometric Identites
   • ∫1/a+b sin^2 x+ cos^2 x dx
     • Sub t =tan x
   • ∫1/a+b sin x+c cos x dx
     • Sub t=tan x/2

4. 4. Try again.

   • If everything that you’ve tried to this point doesn’t work
     • Try substitution
     • Try parts
     • Manipulate the integrand
     • Relate the problem to a previous problem
     • Use several methods
       • Completing the Square
         • When have quadratic function at denominator
       • Trigonometric Identities
         • When having different angle
         • When having summation of two term power two
       • Long division Method
         • For improper Fraction
       • Seperating the fraction
       • Multiplying by a Form of 1
         • Multiply The expression by some term divided by itself
       • Eliminating the Square Root
         • When trigonometric Function in the Square Root
       • Trigonometric Substitution
         • When replace the variable of integration by a trigonometric function