5.6 Integration by Substitution Method (U-substitution)

Fri Feb 5Do Now

Find the derivative of

HW Review: p.326

Reverse Chain Rule

• Looking at the 2 Do Now problems, we can say

• Notice how 2 factors integrate into one

Substitution Method

• If F’(x) = f(x), then

Integration by Substitution(U-Substitution)

• 1) Choose an expression for u– Expressions that are “inside” another function

• 2) Compute • 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

Ex 2 – Multiplying du by constant

• Evaluate

Ex 3 – u in the denominator

• Evaluate

Ex 4 - Trig

• Evaluate

Ex 5 – Integrating tangent

• Evaluate

Ex 6 – 2 step Substitution

• Evaluate

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

Closure

• Evaluate the integral

• HW: p.333-335 #1-89 EOO due Monday, 1-89 AOO due Tuesday

5.6 U-Substitution Review / Practice

• Do Now• Evaluate the integrals• 1)

• 2)

HW Review: p.333 1-89

Practice

• Worksheet if time

Closure

• Evaluate the integral

• HW: p.333 #1-89 AOO

5.6 Substitution MethodTues Feb 9

• Evaluate the integral using substitution

HW Review: p.333 1-89

Practice

• Worksheet

Closure

• When do we use substitution when integrating? How does it work? What about with definite integrals?

• HW: none