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Chapter 5: Problem 1
On which derivative rule is the Substitution Rule based?
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Additional integrals Use a change of variables to evaluate the following integrals. $$\int \sec 4 w \tan 4 w d w$$
General results Evaluate the following integrals in which the function \(f\) is unspecified. Note \(f^{(p)}\) 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. \(\int\left(f^{(p)}(x)\right)^{n} f^{(p+1)}(x) d x,\) where \(p\) is a positive integer, \(n \neq-1\)
Additional integrals Use a change of variables to evaluate the following integrals. $$\int \sin x \sec ^{8} x d x$$
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 \((b)\) $$f(x)=e^{x} ; a=0, b=\ln 2, c=\ln 4$$
\(\sin ^{2} a x\) and \(\cos ^{2} a x\) integrals Use the Substitution Rule to prove that $$\begin{array}{l}\int \sin ^{2} a x d x=\frac{x}{2}-\frac{\sin (2 a x)}{4 a}+C \text { and } \\\\\int \cos ^{2} a x d x=\frac{x}{2}+\frac{\sin (2 a x)}{4 a}+C\end{array}$$
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