# 5.2 Integration by Substitution

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```Calculus
5.2
5.2 Integration by Substitution
A. Getting Started
1. We’ll take an integral which we can’t currently
solve and rewrite it using substitution to create an
integral that we can solve.
2. Steps for Using Substitution
a. Find u
b. Calculate du
1) Solve for dx
c. Substitute u and dx into the problem
1) All x should cancel out
d. Integrate
e. Substitute back to x
Calculus
5.2
3. Let u be:
biggest power
a. x
biggest power
then u = x
power
b. e
then u = power
c. (
) power
then u = (
)
)
then u = (
)
?
d. (
4. Three main integrals we’ll end up with:
n +1
u
n
u
a. ∫ du = n + 1 + c
u
u
e
du
=
e
+c
b. ∫
1
c. ∫ u du = ln u + c
Calculus
5.2
5. Calculating du
a. Once we find u, considering taking the derivative
of both sides of the equal sign
b. Find du
4
1) u = x
5
2) u = 8 − x
Calculus
5.2
B. Find the Indicated Integral
1.
∫ 2 xe
∫
x2
dx
2 x3
2. 18 x e dx
Calculus
3.
∫ 4x e
3 x4
2 xdx
4. ∫ x 2 + 1
5.2
dx
Calculus
x 2 dx
5. ∫ 7 + x 3
10 x 4 dx
6. ∫ 6 + x 5
5.2
Calculus
7.
8.
∫ 7t ( t
∫
(m
5.2
2
)
5
− 1 dt
2m − 7
2
− 7m + 3
)
5
dm
Calculus
2
6
q
q
− 1dq
9. ∫
10.
ex
∫ e x − 7dx
5.2
Calculus
11.
12.
∫
∫
5.2
v+5
3
e
v + 10v − 1
2
t
t
dt
dv
Calculus
C. Applications
1. The marginal revenue from the sale of x units is
−0.01 x 2
′
=
+
R
(
x
)
50
3.5
xe
dollars per
estimated to be
unit where R ( x) is the revenue in dollars. Find
R ( x) , where R (0) = 0 .
5.2
Calculus
5.2
2. It is estimated that t weeks from now, the price of
chicken will be increasing at the rate of
p′(t ) = 3 t + 1 cents per kg per week. If chicken
currently costs \$2.30 per kg, what will it cost 8
weeks from now?
```