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1 5.5 – The Substitution Rule

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2 Example – Optional for Pattern Learners 1. Evaluate 3. Evaluate Use WolframAlpha to evaluate the following. 2. Evaluate Notice that each of these are of the form where u is some function of x. If is the antiderivative of f is F, what will be the answer of an indefinite integral of this form?

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3 The Substitution Rule – The Idea Evaluate Clearly, the answer is e sin x + c. How do we arrive at this? Let u = sin x. Then du/dx = cos x. Substituting this into the integral above, we get This is a simple integral that we can evaluate and get e u + c. Substituting sin x in for u, we have our answer. \ \

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4 The Substitution Rule If u = g(x) is a differentiable function whose range is an interval I and f is continuous on I, then Since, u du

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5 The Substitution Rule – General Technique 1.Let u be g(x) in the composition. You may wish to rearrange the function so that anything not in the composition is in front of the dx. 2.Determine du/dx and multiply both sides by dx. 3.Divide both sides by the constant, if necessary. Note: There are variations on this technique so it may have subtle changes in process. Also, you may have a different variable than x so us du/d[variable] in step 2.

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6 Examples - Evaluate

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7 Substitution Rule Twists - Examples Sometimes things are not as obvious and it may not seem that you can find a u = g(x) such that du = g'(x) dx. With a little creativity, you can. You may need to solve for x in terms of u. Evaluate the following. This uses the standard substitution technique, but has a little twist.

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8 The Substitution Rule – Definite Integrals If u = g(x) is a differentiable function whose range is an interval I and f is continuous on I, then Properties: Suppose f is continuous on [– a, a].

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9 Examples - Evaluate

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