Integration Flashcards
Improper integral
Integral evaluated as b approaches infinty
Main techniques used to solve
Inspection
U-sub
By parts
Partial fractions
Inspection
If numerator is derivative of the denominator then integral is simply ln(denominator) + c
How can the fraction be manipulated so it fits for inspection
We can multiply by n/n and remove some constants from within the integral, alternatively we can use partial fractions or simply splitting the numerator
U Substitution
Aim to simplify integral with the definition of du and get it into a standard integral form that is easily solved.
Integration by parts equation
∫u’v = uv -∫uv’
Integration by parts for cyclic functions
Set integral = to I, eventually our -∫uv’ will equal our original so we resub in I and rearrange to solve.
(May need to show that we’ve changed c)
Alternative approaches related to integration by parts
May have to create a u’ = 1 or alternatively vary which is u’ and which is v as may not always be obvioius/ straightforward
Partial fractions used when:
Need to split up an integral into more easily analysed components:
3 forms of partial fractions
no repeated root, repeated root and irreducible quadratic
no repeated root
= A/(x-n1) + B/(x-n2)
repeated root
= A/(x-n1) + B/(x-n2) + C/(x-n2)^2
quadratric
= A/(x-n1) + (Bx+C)/(x^2-n2x + n3)
How to solve partial fractions
Multiply equation by denominator and use logical values of x to set some constants to 0 and then solve
What if order of numerator is greater than that of the denominator
We use long algebraic divion
Process of long algebraic division
Set up division with denominator as the divisor. Then multiply divisor by values that will help solve the equation when we have numerator - denominator and values will = 0. These values go to the top of the division and as bottom slowly simplifies to a point until it cant be simplified further, then our answer will be top line + remainder/divisor
Length of curve equation
L =∫ (1+(f’(x))^2)^1/2 dx
Integral of parametric equations
L = ∫ ((y(t))^2 + (x(t))^2)^1/2
How to integrate absolute values:
Create a piecewise function that has one part when |inside| when inside is greater than 0 and one where inside is less than 0 (with this one being multiplied by -1)
We can then split the integral and determine what the limits will be then solve normally.
Technique to solve integral with difficult bottom
Multiply by conjugate to try simplify (a^2 - b^2)
