Last week we learned something new, Integration by Substitution. This is, in a way, the product/quotient rule of integration because it allows you to integrate terms that are being multiplied and divided.
To use substitution, one function must be the derivative of another.
*If you are only off by a number, tho, it can be fixed.
Here is an example of substitution:
x ( x^2 + 1 )^3
-First, find your "u" and your "du"
u = x^2 + 1 du = 2x
The problem has to become 1/2 S 2x ( x^2 + 1 )^3 because:
du = 2x (du is the derivative of u) that means that on the front side of the integration symbol, you must put the reciprical of 2. This is to balance the eqn out because you had to add a 2 to the problem.
-Now replace the terms with u and du:
1/2 S u^3 du
-Now integrate:
1/2 (1/4u^4) + c
-Now plug in for u and simplify and you get:
1/8 (x^2 + 1)^4 + c
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