SECTION 6.2Integration by Substitution

U-SUBSTITUTION

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

+ C

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.

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.

WHAT IF THERE IS A CONSTANT LEFT OVER?

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

=

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.

ADDITIONAL EXAMPLES

1.

2.

3. Challenge:

USING TRIG IDENTITIES

Evaluate the following integrals by using trig identities first.

1.

2.

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).

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.

EXAMPLE

Calculate the integral using both methods…

ADDITIONAL EXAMPLES

1. Evaluate

2. Evaluate

AP MC PRACTICE

AP MC PRACTICE

MORE AP MC PRACTICE

DO-NOW: HOMEWORK QUIZ

Evaluate the following indefinite integrals.1.

2.

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).

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.