Chapter 6: Problem 1
Write the form of the partial fraction decomposition of the rational expression. Do not solve for the constants. $$ \frac{5}{x^{2}-10 x} $$
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Determine all values of \(p\) for which the improper integral converges. $$ \int_{1}^{\infty} \frac{1}{x^{p}} d x $$
Use integration by parts to verify the reduction formula. $$ \int \sin ^{n} x d x=-\frac{\sin ^{n-1} x \cos x}{n}+\frac{n-1}{n} \int \sin ^{n-2} x d x $$
Laplace Transforms Let \(f(t)\) be a function defined for all positive values of \(t\). The Laplace Transform of \(f(t)\) is defined by \(F(s)=\int_{0}^{\infty} e^{-s t} f(t) d t\) if the improper integral exists. Laplace Transforms are used to solve differential equations. Find the Laplace Transform of the function. $$ f(t)=\sin a t $$
In Exercises 65 and 66, apply the Extended Mean Value Theorem to the functions \(f\) and \(g\) on the given interval. Find all values \(c\) in the interval \((a, b)\) such that \(\frac{f^{\prime}(c)}{g^{\prime}(c)}=\frac{f(b)-f(a)}{g(b)-g(a)}\) \(\begin{array}{l} \underline{\text { Functions }} \\ f(x)=\sin x, \quad g(x)=\cos x \end{array} \quad \frac{\text { Interval }}{\left[0, \frac{\pi}{2}\right]}\)
Use a graphing utility to graph \(f(x)=\frac{x^{k}-1}{k}\) for \(k=1,0.1\), and 0.01 . Then evaluate the limit \(\lim _{k \rightarrow 0^{+}} \frac{x^{k}-1}{k}\).
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