91Ó°ÊÓ

a. Multiply the numerator and denominator of sec \(x\) by \(\sec x+\tan x ;\) then use a change of variables to show that $$\int \sec x d x=\ln |\sec x+\tan x|+C$$ b. Use a change of variables to show that $$\int \csc x d x=-\ln |\csc x+\cot x|+C$$

Evaluate the following definite integrals using the Fundamental Theorem of Calculus. $$\int_{1}^{8} \sqrt[3]{y} d y$$

Evaluate the following definite integrals using the Fundamental Theorem of Calculus. $$\int_{\sqrt{2}}^{2} \frac{d x}{x \sqrt{x^{2}-1}}$$

Evaluate the following definite integrals using the Fundamental Theorem of Calculus. $$\int_{1}^{2} \frac{z^{2}+4}{z} d z$$

Evaluate the following integrals in which the function \(f\) is unspecified. Note that \(f^{(p)}\) is the pth derivative of \(f\) and \(f^{p}\) is the pth power of \(f .\) Assume \(f\) and its derivatives are continuous for all real numbers. $$\int\left(5 f^{3}(x)+7 f^{2}(x)+f(x)\right) f^{\prime}(x) d x$$

Evaluate the following definite integrals using the Fundamental Theorem of Calculus. $$\int_{0}^{\sqrt{3}} \frac{3 d x}{9+x^{2}}$$

Evaluate the following integrals in which the function \(f\) is unspecified. Note that \(f^{(p)}\) is the pth derivative of \(f\) and \(f^{p}\) is the pth power of \(f .\) Assume \(f\) and its derivatives are continuous for all real numbers. $$\begin{aligned} &\int_{1}^{2}\left(5 f^{3}(x)+7 f^{2}(x)+f(x)\right) f^{\prime}(x) d x, \text { where } f(1)=4\\\ &f(2)=5 \end{aligned}$$

Find the area of the region \(R\) bounded by the graph of \(f\) and the \(x\) -axis on the given interval. Graph \(f\) and show the region \(R\) $$f(x)=2-|x| \text { on }[-2,4]$$

Evaluate the following integrals in which the function \(f\) is unspecified. Note that \(f^{(p)}\) is the pth derivative of \(f\) and \(f^{p}\) is the pth power of \(f .\) Assume \(f\) and its derivatives are continuous for all real numbers. $$\int_{0}^{1} f^{\prime}(x) f^{\prime \prime}(x) d x, \text { where } f^{\prime}(0)=3 \text { and } f^{\prime}(1)=2$$

Find the area of the region \(R\) bounded by the graph of \(f\) and the \(x\) -axis on the given interval. Graph \(f\) and show the region \(R\) $$f(x)=\left(1-x^{2}\right)^{-1 / 2} \text { on }[-1 / 2, \sqrt{3} / 2]$$