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Chapter 5: Problem 6
If the change of variables \(u=x^{2}-4\) is used to evaluate the definite integral \(\int_{2}^{4} f(x) d x,\) what are the new limits of integration?
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Use geometry and the result of Exercise 76 to evaluate the following integrals. $$\int_{1}^{6} f(x) d x, \text { where } f(x)=\left\\{\begin{array}{ll}2 x & \text { if } 1 \leq x<4 \\\10-2 x & \text { if } 4 \leq x \leq 6\end{array}\right.$$
Additional integrals Use a change of variables to evaluate the following integrals. $$\int\left(x^{3 / 2}+8\right)^{5} \sqrt{x} d x$$
Show that the Fresnel integral \(S(x)=\int_{0}^{x} \sin \left(t^{2}\right) d t\) satisfies the (differential) equation \(\left(S^{\prime}(x)\right)^{2}+\left(\frac{S^{\prime \prime}(x)}{2 x}\right)^{2}=1\)
Consider the function \(f\) and the points \(a, b,\) and \(c\) a. Find the area function \(A(x)=\int_{a}^{x} f(t) d t\) using the Fundamental Theorem. b. Graph \(f\) and \(A\) c. Evaluate \(A(b)\) and \(A(c)\) and interpret the results using the graphs of part \((b)\) $$f(x)=e^{x} ; a=0, b=\ln 2, c=\ln 4$$
Evaluate the following definite integrals using the Fundamental Theorem of Calculus. $$\int_{1}^{2} \frac{z^{2}+4}{z} d z$$
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