Math 142 Lecture Notes
Section 6.2 – Integration by substitution
A) Differentials
Definition
If y = f(x) defines a differentiable function, then
1.- The differential dx of the independent variable is an arbitrary real number
2.- The differential dy of the dependent variable is defined as dy = f '(x) dx .
Examples
1) y = x2 => dy = 2x dx
2) y = e3x => dy = 3e3x dx
3) y = ln( 4 + 5x ) => dy = 5 / (4 + 5x) dx
B) Substitution Method
Let's start with the following
Examples
1)
∫  x1 dx
2)
∫ 2x  x2 1 dx
3)
∫ 4x3  x47 1 /3 dx
The Substitution Rule
If u = g(x) is a differentiable function whose range is an interval I and f is continuous on I, then
∫ f g x g'  x dx=∫ f u du
Thus, we have a method for some integrals where the integrand can be transformed into
another one for which we know the way to solve it
Examples
1)
∫ 8  8x1 dx
2)
∫ 6x  3x2 −12dx
3)
∫ 2x3− x 4 dx
2
6x −1
C) General Indefinite Integrals
Determine if the function F(x) is the general anitderivative of the function f(x).
n
n1
u
C
n1
a)
∫ u du=
b)
∫ eu du=euC ; u = u(x)
c)
1
∫ u du=ln∣u∣C ; u = u(x)
; n ≠ -1 ; u = u(x)
Examples
1)
∫ 3x 4−7x 212x3−14x  dx
2)
∫ e5 x 15 x2 dx
3)
∫ 5x3− 4x2124 dx
4)
∫ x3  x 4−14dx
5)
∫ 10x10x 2 2x−73 dx
6)
∫ 2x3−54 dx
7)
∫ e−5x dx
8)
∫
3
2
15x −8x
x
2
12x
dx
2
4
2x −6
C) Solving for x in the equation u = u(x)
Examples
1)
∫ x  x −2 dx
2)
∫ x7  x4−2 dx
3)
∫
x
 x5
dx