Calculus 6.2

Calculus-IIMath-102Dr. Farhana Shaheen5.3 Integration by substitution

Always Remember:Integration is by Observation And The Best way is by Substitution

Integrals of Transcendental FunctionsTrigonometric functionsThe inverse trigonometric functionsExponential functionsLogarithms.

INTEGRATION BY SUBSTITUTION

Note: Integration by substitution can be used for a variety of integrals: some compound functions, some products and some quotients.

Sometimes we have a choice of method.

Choosing u

Guidelines for u-Substitution (p. 334)

(Page: 337)Equation (5)

Equation (6) Equation (7)

The chain rule allows us to differentiate a wide variety of functions, but we are able to find antiderivatives for only a limited range of functions? We can sometimes use substitution or change of variable to rewrite functions in a form that we can integrate.

Example 1:

The variable of integration must match the variable in the expression.

Dont forget to substitute the value for u back into the problem!

Example 2:

One of the clues that we look for is if we can find a function and its derivative in the integral.

The derivative of is .

Note that this only worked because of the 2x in the original.Many integrals can not be done by substitution.

Example 3:

Solve for dx.

Example 4:

Example 5:

We solve for because we can find it in the integral.

Example 6:

Example: 7

Example: 8

Example: 10 (page 336) Evaluate:

Example: 11 (page 336) Evaluate:

Exercises

Use substitution to integrate the following. (Where possible, you could also use a 2nd method.)

1.

2.

3.

Question for you:

Ex: 5.3 Questions: 1 54 (page 338)

Questions: 1 12 (substitution is given)Questions: 15 54 (using substitution)

Questions: 61 62 (using substitution and formulas)