Blog #31

I’m going to review how to integrate using substitution. This is used when you have a product or quotient rule. This can’t always be used, but these are the instances in which it can:
S (u)(v)
…or…
S u/v
where v is the derivative of u

Here’s an example:
Given…S (x²+1)(2x) dx
First, identify u and v. Which is the derivative and which is the original?
u = x²+1
v = 2x
…because 2x is the derivative of x²+1…
Okay, next you remove v (2x dx) from the equation and substitute “u” for x²+1.
S u du
*You insert du because v is the derivative of u, hence du.
Now, integrate u.
½ u² + C
Finally, replace “u” with the original.
½ (x²+1)² + C

I’m also going to review the Second Fundamental Theorem of Calculus:
This theorem state that the derivative of an integral results in the original equation.
The formula: d/dx vSa f(x) dx = (f(v)) (v’) where “a” is a number (usually equaling 0) and v is an equation

An example:
d/dx x²S0 (x+1)² dx
= (((x²) +1)²) (2x)
= 2x(x²+1)²