Different ways to attack an integral

Different ways to attack an integral

1. Facts

   • There are different ways to attack an integral, the attack depends on the enemy, you must attack the integral depending on its type, to do this you must identify its characteristics
   • Some integrals require of algebraic process

2. Direct (basic)

   • The basic integral is defeated by using the basic integration formula
   • Formula
     • x^n+1/n+1
   • How to identify it
     • This integral is identified due to its simplicity

3. Substition

   • Substitution with u
     • How to identify it
       • These are identified by identifying a posible derivate that might substitute something in the integral applying u
     • Example
     • To defeat it you must identify a value that derivated cancels out a value of the integral, substitute that value for u and then derivate
   • Change of variables
     • Solution
     • To defeat it you must do the same as the u substitution with the difference that the remaining are going to fit in the ecuation with its reciplocal
     • How to identify it
       • Same as u substitution, but don't search for the whole canceling but part of it

4. Trigonometric integrals

   • Positive odd Sin and positive odd cosine
     • How to identify it
       • We can identify these problems by recognizing the presence of a sine or cosine
     • Problem
       • This problem is solved similar to the substition method, with the only exception of the trigonometrical funtions of sin and cosine being derivated as a trigonometrical function when finding u and du following to the substition procedure as normal integrating according to the trigonometric integrals listened above
     • There are different complex cases which examples are listed here:
   • Facts
     • There are a lot of cases in these problems which involve the use of trigonometric identities, trigonometric derivates, etc. this is due to the fact that some of these derivates need the use of the separation of power and the knowledge of different ways to solve the integral
   • Tan and Sec
     • To integrate with tan and sec you must use the the trigonometrical identities to obtain sin and cosine, so you can derivate with substition using sin and cosine
     • Identities
   • Here are the integrals of the trigonometric functions after the susbstitution is done
   • Complex trigonometric functions

5. Partial Fractions

   • Inegration by partial fractions is basically using algebra to make polinomial fractions simplier to solve
   • Steps
     • Verify that the denominator has a bigger value than the numerator
     • Identify the factors of the denominator
     • According to the listed table, separate the partial fraction terms based on the denominator
     • Multiply the denominator values by the values of the whole ecuation
     • Simplify and obtain the values of each of the values
     • Solve the integral with the given values by using one of the integration methods

6. Integration by parts

   • Use of formula
     • Formula
     • Example
     • To solve these problems you need to use the formula, you obtain these values by using LIPET arranging the stuff in the correct way
       • LIPET
         • The word “LIPET” is an acronym, meaning that each letter stands for a word. In this case, the letters represent different types of functions. These identifications are: L = Logarithmic function I = Inverse trigonometric function P = Polynomial function E = Exponential function T = Trigonometric function
     • This is used when there is an ln as it is the only way to solve it (Integration by parts)
   • Cycle
     • These can be solved with the formula but it is way harder
     • Derivate the u and integrate the dv using lipet
       • LIPET

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