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In the following exercises, verify each identity using differentiation. Then, using the indicated $u$ -substitution, identify $f$ such that the integral takes the form $\int f(u) d u .$$$\int x \sqrt{4 x^{2}+9} d x=\frac{1}{12}\left(4 x^{2}+9\right)^{3 / 2}+C ; u=4 x^{2}+9$$
Calculus 1 / AB
Calculus 2 / BC
Chapter 1
Integration
Section 5
Substitution
Integrals
Integration Techniques
Campbell University
Harvey Mudd College
University of Michigan - Ann Arbor
Lectures
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The first thing we can do is after differentiating both sides with respect to X, we can apply the general so F five acts. In other words, the derivative equals 1/12 times three over two times four X squared, plus nine to the three over two minus one D over detox tips for X squared plus nine plus de over Jack's Times See the constant Okay, now we some Now we can simplify this to be f crime of acts equals four x squared plus not into the 1/2 times acts or I personally prefer to write it with radicals four X squared plus nine x times for swordplay is mine. Okay, now we know that are you is for X squared plus nine r D u is eight x d x and we know that one. It do you equals x d X. Therefore, we have to pull out the 1/8 because remember, we pull out the constant more doing intervals. So we write it on the outside. So this is the integrate sign, and we write 1/8 on the outside. It's the constant, and then we have the square root of you d'you because remember you equals four X cripples nines. The skirt of forks cripples nine. So this is how you would write it in the form.
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