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INTEGRATION BY SUBSTITUTION

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Substitution with Indefinite Integration This is the “backwards” version of the chain rule Recall … Then …

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Substitution with Indefinite Integration In general we look at the f(x) and “split” it –into a g(u) and a du/dx So that …

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Substitution with Indefinite Integration Note the parts of the integral from our example

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Substitution with Indefinite Integration Let u = So, du = (2x -4)dx

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Guidelines for Making a Change of Variables

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Theorem 4.13 The General Power Rule for Integration

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Example 1: The variable of integration must match the variable in the expression. Don’t forget to substitute the value for u back into the problem!

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Guildelines If something is being raised to an exponent (including a radical), that will be u. If one function is 1 degree higher than the other function, that will be u. If e is being raised to an exponent, that exponent will be u. If you have one trig function, the inside function will be u.

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Example 2: One of the clues that we look for is if we can find a function and its derivative in the integrand. The derivative of is.Note that this only worked because of the 2x in the original. Many integrals can not be done by substitution.

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Example 3: Solve for dx.

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Example 4:

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Example 5: We solve for because we can find it in the integrand.

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Example 6:

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Can You Tell? Which one needs substitution for integration?

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Integration by Substitution

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Solve the differential equation

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Theorem 4.14 Change of Variables for Definite Integrals

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or you could convert the bound to u’s.

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Example 7: The technique is a little different for definite integrals. We can find new limits, and then we don’t have to substitute back. new limit

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Example 9: Don’t forget to use the new limits.

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Theorem 4.15 Integration of Even and Odd Functions

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Even/Odd Functions If f(x) is an even function, then If f(x) is an odd function, then

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Even/Odd Functions If f(x) is an even function, then If f(x) is an odd function, then

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