Warm Up

Sect. 4.5 Integration By Substitution

Chain Rule for Derivatives: Chain Rule backwards for Integration: This means that in order to integrate a composite function, the derivative of the “inside” must be present, and you integrate the “outside.”

Back to Warm Up 1 Let u = 3x – 2x2, then du/dx = 3 – 4x

Back to Warm Up 2 du Let u = 1 – 2x2, then du/dx = – 4x du = – 4x dx

u – substitution in Integration Let g be a function whose range is an interval I, and let f be a function that is continuous on I. If g is differentiable on its domain and F is an antiderivative of f on I, then

Recognizing the Pattern Given each integral, determine u = g(x) and du = g’(x)dx. g’(x) g(x) g’(x) (g(x))1/2 (g(x))2 g(x) g’ g(x) g’(x) g(x) eg(x) cos(g(x))

Making a Change of Variables The steps used for integration by substitution are summarized in the following guidelines. 1. Choose a substitution u = g(x). Usually it is best to choose the inner part of a composite function. 2. Compute du = g’(x) dx 3. Rewrite the integral in terms of the variable u. 4. Evaluate the resulting integral in terms of u. 5. Replace u by g(x) to obtain an antiderivative in terms of x. 6. Check your answer by differentiating.

Remember you may differentiate to check your work! Ex 1: u = x2 + 1 Remember you may differentiate to check your work! Substitute, simplify and integrate:

Ex 2: u = 5x Substitute, simplify and integrate:

Ex 3: Find . Let Substitute, simplify and integrate:

Ex 4: Find . Substitute into the integral, simplify and integrate.

You Try… Evaluate each of the following.

Example 1 Solution u1 du u = x2 + x u2/2 Example 3 Solution

Multiplying/Dividing by a Constant Ex 5: Let u = x2 + 1 du = 2x dx problem! Rewrite original with factor of 2

Ex 6:

Warning! The constant multiple rule only applies to constants!!!!! You CANNOT multiply and divide by a variable and then move the variable outside the integral.

You Try… Evaluate each of the following. 6.

Example 4 Solution Let u = 4x3 then du = 12x2 dx Example 5 Solution Let u = 5 – 2x2 then du = - 4x dx

Example 6 Solution

More Complicated Examples u = sin 3x du = 3cos 3x

Example 8 Evaluate

You Try… Evaluate each of the following.

Example 1 Solution

Example 9 Find

Example 10

Example 11

You Try… Evaluate each of the following.

Example 6 Solution

Example 12

Example 13

Example 14

Example 15

Warm Up Evaluate

Shortcuts: Integrals of Expressions Involving ax + b

Back to Warm Up Evaluate

Evaluate each of the following. 1. 2. 3. You Try… Evaluate each of the following. 1. 2. 3.

For some integrals, the shortcut does not apply For some integrals, the shortcut does not apply. Instead, FOIL and integrate using the Power Rule. 

Definite Integrals

Ex 1: u = x2 + 1 du = 2x dx Or, we could use new limits if we leave u in. Calculate new upper and lower limits by substituting the old ones in for x in u. x = 0  u = 1 x = 1  u = 2 Notice

You Try… Sketch the region represented by each definite integral then evaluate. Use a calculator to verify your answer. 1. 2.

One option is to change the limits 1 Solution One option is to change the limits u = 3t - 1 Then when t = 1, u = 2 when t = 2, u = 5 Resulting integral

2 Solution Don’t forget to use the new limits.

Ex 2:

Using the original limits: Ex 2: Using the original limits: Leave the limits out until you substitute back.

Ex 2: Using the new limits: new limit new limit

You Try… Sketch the region represented by each definite integral then evaluate. Use a calculator to verify your answer. 3.

3 Solution

You Try… 4.

Definite Integrals & Transformations Given , find a. b. c. d. e. f.

You Try… Given , find a. b. c. d. e. f.

Even/Odd Functions Given the graph, evaluate the definite integral of . Given the graph and , evaluate .

Even – Odd Property of Integrals

Symmetric about the y axis FUNCTIONS Symmetric about the origin

A function is even if f( -x) = f(x) for every number x in the domain. So if you plug a –x into the function and you get the original function back again it is even. Is this function even? YES Is this function even? NO

A function is odd if f( -x) = - f(x) for every number x in the domain. So if you plug a –x into the function and you get the negative of the function back again (all terms change signs) it is odd. Is this function odd? NO Is this function odd? YES

If a function is not even or odd we just say neither (meaning neither even nor odd) Determine if the following functions are even, odd or neither. Not the original and all terms didn’t change signs, so NEITHER.

Ex 3: a) Evaluate . Find the area of the region bounded by the x-axis and the curve .

Example 4: Evaluate Let f(x) = x4 – 29x2 + 100 Since f(x) = f(-x), f is an even function.

Since f(x) = f(-x), f is an odd function. Ex 5: Evaluate Let f(x) = sin3x cos x + sin x cos x f(–x) = sin3(–x) cos (–x) + sin (–x) cos (–x) = –sin3x cos x – sin x cos x = –f(x) Since f(x) = f(-x), f is an odd function.

You Try… Find the area of the region bounded by the x-axis and the curve of on the interval [-2, 2].

You Try… -a a

u-sub with a Twist Ex 1: problem! Substitute in terms of u Antiderivative in terms of u Antiderivative in terms of x

Ex 2: problem! Substitute in terms of u Antiderivative in terms of u Antiderivative in terms of x

Alternate u Ex 2:

Ex 3: Evaluate

You Try…

Example 2 Solution u = 2x + 3 du = 2 dx

determine new upper and lower limits of integration Back to Ex 2: as a definite integral determine new upper and lower limits of integration When x = 1, u = 1 lower limit upper limit x = 5, u = 9

Ex 2: (cont.) Before substitution After substitution = Area of region is 16/3 Area of region is 16/3

Example 4

You Try…

Example 3 Solution u = x + 1 du = dx

Using Long Division Before Integrating Back to Example 3

Example 5

You Try…

Evaluate each of the following. 1. 2. Warm Up Evaluate each of the following. 1. 2.

Integrals of Trig Functions A useful example of integration by substitution is to find Ex 1: Let Using a law of logs,

A little trickier one… First, multiply numerator and denominator by sec x + tan x. u = sec x + tan x, du = (sec x tan x + sec2x) ∫ (1/u) du = ln |u| + C Ex 2:

Integrals of Trig Functions

Shortcuts: Integrals involving (ax + b)

Examples: 3. 4. Evaluate

Sect. 5.9 Inverse Trig Functions: Integration

Integrals of Inverse Trig Functions If each integral sign above had a – in front, we would get an antiderivative of the corresponding co-function.

Ex 1 Ex 2

Ex 3 Ex 4

You Try… Evaluate each of the following.

Ex 1 Solution

Ex 5

You Try… Evaluate each of the following. Hint: Split up into two separate fractions first.

You Try… 5. Find the area enclosed by x = -¼, x = ¼, y = 0, and

Warning Many integrals look like the inverse trig forms, but require a different technique. Which of the following are of the inverse trig forms? If they are not, how are they integrated?