5.6 Integration by Substitution
Method (U-substitution)
Thurs Dec 3
Do Now
Find the derivative of
f (x) = sin(x )
2
g(x) = sin(x + x 3 )
HW Review
Reverse Chain Rule
• Looking at the 2 Do Now problems, we can say
2
2
2x
cos(x
)
dx
=
sin(x
)
ò
ò (1+ 3x )cos(x + x ) dx = sin(x + x )
2
3
3
• Notice how 2 factors integrate into one
Substitution Method
• If F’(x) = f(x), then
ò f (u(x))u'(x) dx = F(u(x))+ C
Integration by Substitution
(U-Substitution)
• 1) Choose an expression for u
– Expressions that are “inside” another function
du
• 2) Compute du = dx
dx
• 3) Replace all x terms in the original integrand
so there are only u’s
• 4) Evaluate the resulting (u) integral
• 5) Replace u after integration
Expressions for U-substitution
•
•
•
•
Under an exponent
Inside a function (trig, exponential, ln)
In the denominator
The factor in a product with the higher exponent
• Remember: you want to choose a U expression
whose derivative will allow you to substitute the
remainder of the integrand!
Ex1
• Evaluate
ò 3x
2
sin(x ) dx
3
Ex 2 – Multiplying du by constant
• Evaluate
ò x(x
2
+ 9) dx
5
Ex 3 – u in the denominator
• Evaluate
ò
(x + 2x)
dx
3
2
6
(x + 3x +12)
2
Ex 4 - Trig
• Evaluate
ò sin(7q + 5) dq
Ex 5 – Integrating tangent
• Evaluate
ò tan x dx
Ex 6 – 2 step Substitution
• Evaluate
òx
5x +1 dx
Substitution and Definite Integrals
• When using u-substitution with definite
integrals you have 2 options
– Plug x back in and evaluate the bounds that way
– Change the x bounds into u bounds and evaluate
in terms of u
Ex
• Evaluate
ò
2
0
x 2 x 3 +1 dx
Closure
• Evaluate the integral
òx
2
cos(x +1) dx
3
• HW: p.333-335 #9 13 20 31 43 53 59 67 69 74
81 89 95
5.6 U-Substitution Review / Practice
Wed Feb 11
• Do Now
• Evaluate the integrals
8
• 1)
ò sin x cos x dx
• 2)
ò
2x 2 + x
dx
3
2 2
(4x + 3x )