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Chapter 5: Problem 1
On which derivative rule is the Substitution Rule based?
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Substitutions Suppose that \(f\) is an even function with \(\int_{0}^{8} f(x) d x=9 .\) Evaluate each integral. a. \(\int_{-1}^{1} x f\left(x^{2}\right) d x\) b. \(\int_{-2}^{2} x^{2} f\left(x^{3}\right) d x\)
Substitutions Suppose that \(p\) is a nonzero real number and \(f\) is an odd integrable function with \(\int_{0}^{1} f(x) d x=\pi .\) Evaluate each integral. a. \(\int_{0}^{\pi /(2 p)} \cos p x f(\sin p x) d x\) b. \(\int_{-\pi / 2}^{\pi / 2} \cos x f(\sin x) d x\)
Use a change of variables to evaluate the following integrals. $$\int_{-1}^{1}(x-1)\left(x^{2}-2 x\right)^{7} d x$$
Use a change of variables to evaluate the following integrals. $$\int \frac{\csc ^{2} x}{\cot ^{3} x} d x$$
If necessary, use two or more substitutions to find the following integrals. \(\int \frac{d x}{\sqrt{1+\sqrt{1+x}}}(\text {Hint: Begin with } u=\sqrt{1+x} .)\)
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