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Published byGarey Shaw Modified over 5 years ago

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6.2 Integration by Substitution & Separable Differential Equations

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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 to rewrite functions in a form that we can integrate.

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Example: 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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Example: 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: Solve for dx.

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Example:

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

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Example:

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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 We could have substituted back and used the original limits.

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Example: Wrong! The limits don’t match! Using the original limits: Leave the limits out until you substitute back. This is usually more work than finding new limits

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

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