Integration and Substitution

Today I have a simple question of integration by substitution. Although it's simple, it is a question many people find difficult.

Integrate (5x + 10)/(3x² + 12x - 7) by substitution

Unlike most short questions, this one also has a short answer. First, we need to define a variable to be one part of the function. This part, when differentiated, must be divisible by another part, or we'll be left with two variables, which is not fun.

As you can see, when you differentiate the denominator, you get 6x + 12, which is similar to the numerator, 5x + 10 (take 6 and 5 as a common factor, respectively). Now all we need is to define a variable and we're set to go:
z = 3x² + 12x - 7
z' = 6x + 12

This can also be written as:
dz/dx = 6x + 12

Make dx the subject:
dx = dz/(6x + 12)

We want to find:
integral((5x + 10)/(3x² + 12x - 7) dx)

Substitute z for the denominator and also replace dx with what we found:
integral((5x + 10)/z dz/(6x + 12))

Take out common factors:
integral(5(x + 2)/z * dz/6(x + 2))

Cancel (x + 2):
integral(5/6z dz)
5/6 * ln(z) + c

Now substitute back the value of z:
5/6 * ln(3x² + 12x - 7) + c

And that's the integral.

Simple, yet somewhat long, isn't it?
Yours,
Nadav

nadavs