Math 152 – Spring 2016
Section 6.5
1 of 4
Section 6.5– The Substitution Rule
Recall, the chain rule from calc I.
d
f (g(x)) =
dx
What is the antiderivative of
Z
f 0 (g(x))g 0 (x) dx =
The goal of substitution is identifying when the integrand is of the correct form f 0 (g(x))g 0 (x)
so that the antiderivative is f (g(x)).
Theorem (The Substitution Rule). If u = g(x) is a differentiable function whose range
is the interval I and f is continuous on I, then
Z
Z
f (g(x))g 0 (x) dx = f (u) du
Note. When you substitute don’t forget three things:
1. You must solve for du and the integral must CONTAIN du. Don’t forget the dx
is part of du.
2. Every variable, usually x, in the integral must be substituted for. You cannot
have an integral with x and u.
3. Check your answer by differentiation with the chain rule.
Example 1. Solve the following integrals with u-substitution.
R
(a) (x5 − 3x2 + 4)7 (5x4 − 6x) dx
R
(b) (4x − 3) sin(2x2 − 6x) dx
Math 152 – Spring 2016
3
x
√
3 2
x −7
(c)
R
(d)
R
tan x dx
(e)
R
√1+x
1−x2
Section 6.5
2 of 4
dx
dx
U-Substitution Tips: (These are just suggestions to try; they are not comprehensive.)
1. If there is a function raised to a power, set u equal to the function (Roots can be
rewritten as powers)
2. If there is a function inside a ln, e, trig. function (sin, etc.), set u equal to the
function.
3. For fractions, try setting the denominator as u. Also, try splitting the fraction
a
b
into multiple fractions, i.e., a+b
c = c + c.
4. Try rewritting trig functions using trig identities.
5. Set u equal to any function whose derivative is also part of the integral (See 2b)
6. Don’t forget the functions whose antiderivatives are tan−1 and sin−1 !
Substitution for Definite Integrals
Theorem (The Substitution Rule for Definite Integrals). If u = g(x), g 0 (x) is continuous on [a, b], and f is continuous on the range of g, then
Z
a
b
f (g(x))g 0 (x) dx =
Z
g(b)
f (u) du
g(a)
Note. When you do a u substitution, you MUST change the limits of integration when
you substitute!
You can change back to x and “x” limits, but NEVER mix “x” limits with u functions.
Math 152 – Spring 2016
Section 6.5
Example 2. Solve the following definite integrals
4
R3
(a) 2 (2x − 3) 3x2 − 9x dx
(b)
R e4 ln x
(c)
R ln a2
(d)
R π/3
1
ln a
0
x
dx
ex
e2x +1
dx (assume a > 0)
sin3 θ dθ
3 of 4
Math 152 – Spring 2016
(e)
R4
0
√ x
2x+1
Section 6.5
4 of 4
dx
Symmetric Functions
Theorem. (Integrals of Symmetric Functions) Suppose f is continuous on [−a, a].
(a) If f is even, i.e., f (−x) = f (x), then
Z a
Z a
f (x) dx = 2
f (x) dx
−a
0
(b) If f is odd, i.e., f (−x) = −f (x), then
Z a
f (x) dx = 0
−a
Example 3. Solve the following integrals.
R3
(a) −3 cos(x5 )x4 dx
(b)
R 10
5 sin x
−10 ex2 −5x2