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Answer the following questions by using the integration technique known as u-substitution.
Joe Foster. u-Substitution. Recall the substitution rule from MATH 141 (see page 241 in the textbook). Theorem If u = g(x) is a differentiable function whose range is an interval I and f is continuous on I, then. ˆ f(g(x))g′(x) dx = ˆ f(u) du. you see why?) Let’s look at . Example 1 Find ˆ sec2(5x + 1) 5 · dx. ˆ sec2(5x + 1) u = 5x + 1. du = 5 dx.
22 kwi 2024 · Paul Seeburger (Monroe Community College) added problems #24 - 28 in 5.5 and #28 - 29, and 31 in 5.6. 5.5E and 5.6E u-Substitution Exercises is shared under a CC BY-NC-SA license and was authored, remixed, and/or curated by LibreTexts. These are homework exercises to accompany Chapter 5 of OpenStax's "Calculus" Textmap.
Identifying which function to take as 'u' simply comes with experience. Some integrals like sin (x)cos (x)dx have an easy u-substitution (u = sin (x) or cos (x)) as the 'u' and the derivative are explicitly given. Some like 1/sqrt (x - 9) require a trigonometric ratio to be 'u'.
x = 4-u. Substitute into the original problem, replacing all forms of x, getting . Click HERE to return to the list of problems. SOLUTION 15 : Integrate . Let u = 2x+3 so that du = 2 dx, or (1/2) du = dx. In addition, we can "back substitute" with x = (1/2)(u-3) . Substitute into the original problem, replacing all forms of x, getting .
The following problems involve the method of u-substitution. It is a method for finding antiderivatives. We will assume knowledge of the following well-known, basic indefinite integral formulas : , where a is a constant. , where k is a constant.
L EMGaGduez dwJi]twhl xIwnEfLivnristQew vC[a^lNckurlIuNsS. Answers to U-substitution Indefinite Integrals #2. 1. (3x2. + 4)4. 1) +. C. 4.
