section 6.2 integration by substitution. u-substitution

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SECTION 6.2 Integration by Substitution

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Page 1: SECTION 6.2 Integration by Substitution. U-SUBSTITUTION

SECTION 6.2Integration by Substitution

Page 2: SECTION 6.2 Integration by Substitution. U-SUBSTITUTION

U-SUBSTITUTION

) = ???) = If we reverse this use of the chain rule, we can write…

+ C

Page 3: SECTION 6.2 Integration by Substitution. U-SUBSTITUTION

USING THE SUBSTITUTION METHOD

We can use this method of changing variables to turn an unfamiliar integral into one that we can work with.

The goal is to replace one portion of the integral with u, and the remainder with . To do the u-substitution method successfully, only constants can remain unaccounted for. Any variable (often x) must be switched to u.

Page 4: SECTION 6.2 Integration by Substitution. U-SUBSTITUTION

EXAMPLE

Let u = sin x. Then du/dx = cos x.Now, either …1. Solve for dx, substitute, and divide out the cos x, or...

2. Recognize that du = cos x dx and make that substitution.

The result: = ……check your answer by taking the derivative.

Page 5: SECTION 6.2 Integration by Substitution. U-SUBSTITUTION

WHAT IF THERE IS A CONSTANT LEFT OVER?

Let u = . Then du/dx = 2x.Method 1: Solve for dx, then substitute: dx = du/(2x).

=

Page 6: SECTION 6.2 Integration by Substitution. U-SUBSTITUTION

METHOD 2: “BEACH BOYS”

Let u = . Then du/dx = 2x.We know that du = 2x dx. We have the x and the dx needed to make the du, but “wouldn’t it be nice” if we had a 2 also.

Multiply the integral by 2 and 1/2. Use the 2 to complete the du, and bring the ½ outside the integral. Then complete the integration as you did before.

Page 7: SECTION 6.2 Integration by Substitution. U-SUBSTITUTION

ADDITIONAL EXAMPLES

1.

2.

3. Challenge:

Page 8: SECTION 6.2 Integration by Substitution. U-SUBSTITUTION

USING TRIG IDENTITIES

Evaluate the following integrals by using trig identities first.

1.

2.

Page 9: SECTION 6.2 Integration by Substitution. U-SUBSTITUTION

DO-NOW: HOMEWORK QUIZ

Suppose that a point moves along some unknown curve y = f(x) in the xy-plane in such a way that at each point (x, y) on the curve, the tangent line has a slope x2. Find an equation for the curve given that it passes through the point (2, 1).

Page 10: SECTION 6.2 Integration by Substitution. U-SUBSTITUTION

U-SUBSTITUTION IN DEFINITE INTEGRALS

Two methods for calculating the definite integral:

1. Perform the u-substitution, integrate, substitute back in for u, and evaluate at the given limits of integration.

2. Perform the u-substitution, integrate, change the limits of integration from x to u, and evaluate the function of u at the new limits of integration.

Page 11: SECTION 6.2 Integration by Substitution. U-SUBSTITUTION

EXAMPLE

Calculate the integral using both methods…

Page 12: SECTION 6.2 Integration by Substitution. U-SUBSTITUTION

ADDITIONAL EXAMPLES

1. Evaluate

2. Evaluate

Page 13: SECTION 6.2 Integration by Substitution. U-SUBSTITUTION

AP MC PRACTICE

Page 14: SECTION 6.2 Integration by Substitution. U-SUBSTITUTION

AP MC PRACTICE

Page 15: SECTION 6.2 Integration by Substitution. U-SUBSTITUTION

MORE AP MC PRACTICE

Page 16: SECTION 6.2 Integration by Substitution. U-SUBSTITUTION

DO-NOW: HOMEWORK QUIZ

Evaluate the following indefinite integrals.1.

2.

Page 17: SECTION 6.2 Integration by Substitution. U-SUBSTITUTION

SEPARABLE DIFFERENTIAL EQUATIONS

A differential equation y’ = f(x, y) is separable if f can be expressed as a product of a function of x and a function of y.

To solve this differential equation….1. Separate the variables : .2. Integrate both sides. The result is an implicit function.

3. Solve for y to get an explicit function (if desired).

Page 18: SECTION 6.2 Integration by Substitution. U-SUBSTITUTION

EXAMPLES

Solve the differential equation:

Solve the initial value problem for the solution you just found if y(0) = 1.

Now try solving some of the differential equations from the slope fields worksheet to see if the solutions match the pictures.