Rules of Integration Flashcards
What do you do whenever you are asked to find the area under a curve?
You integrate the curve equation between the limits provided (if any)
What is integration by (u) substitution also seen as ?
Reverse chain rule
When do you use (u) substitution?
When the integral is in the form:** ∫f[g(x)] * g’(x)**
One of the two multiplying functions** must be a direct derivative/ must contain elements of the derivative **of the other
How do you go about
∫[f’(x)/f(x)] dx
The integral is ln |f(x)| + c
This is a standard (u) substitution that is good to know
What rule of integration do you use for
∫xsin(x) dx
Integration by parts
What is the integration by parts formula?
when an integral is in the form ∫UV’, the formula for the integration is: UV - ∫VU’
where U is one function and V’ is another function
How do we pick U and V’ before performing integration by parts?
Using the (I)LATE formula, logarithms (L) are typically assigned to be U over algebra (A), trig (T) and exponentials (E)
The easiest thing to differentiate is usually made U
What rule of integration do you use for
∫x/[x(x+4)(x-7)] dx
Integration by partial fractions
Can we go about partial fraction decomposition directly below?:
x/[x(x+4)(x-7)]
Yes
You can decompose directly when the power of the variable in the numerator is less than the power of the variable in the denominator
Can we go about partial fraction decomposition directly below?:
(x^2)/[(x+3)(x-3)]
No long division will have to be carried out
You cannot decompose directly when the power of the variable in the numerator is equal to the power of the variable in the denominator
Can we go about partial fraction decomposition directly below?:
[(x^2) + 4x + 12]/[(x+2)]
No, long division will have to be carried out
You cannot decompose directly when the power of the variable in the numerator is greater than the power of the variable in the denominator
Decompose the following into partial fractions:
[(x^2) + 4x + 12]/[(x+1)((x+2)^2)]
A/(x+1) + B/(x+2) + C/((x+2)^2)
When one factor is raised to a power higher than one, it is repeated as many times before it reaches the final power
