Objective: To integrate functions using a u-substitution

U-Substitution The method of substitution can be motivated by examining the chain rule from the viewpoint of antidifferentiation. For this purpose, suppose that F is an antiderivative of f and that g is a differentiable function. The chain rule implies that the derivative of F(g(x)) can be expressed as which we can write in integral form as

U-Substitution For our purposes it will be useful to let u = g(x) and to write in the differential form With this notation, our formula becomes

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Example 1 Evaluate We will let . Most times the function being raised to an exponent will be u.

Example 1 Evaluate We will let . Most times the function being raised to an exponent will be u. If then . We will solve for dx, so .

Example 1 Evaluate We will let . Most times the function being raised to an exponent will be u. If then . We will solve for dx, so . We will now write the integral as

Example 1 Evaluate We will now integrate the new function and then substitute back in for u.

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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