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Chapter 6: Problem 10
Find the integral. (Note: Solve by the simplest method-not all require integration by parts.) $$ \int \frac{e^{1 / t}}{t^{2}} d t $$
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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. $$ f(t)=\cos a t $$
Continuous Functions In Exercises 73 and \(74,\) find the value of \(c\) that makes the function continuous at \(x=0\). \(f(x)=\left\\{\begin{array}{ll}\frac{4 x-2 \sin 2 x}{2 x^{3}}, & x \neq 0 \\\ c, & x=0\end{array}\right.\)
A nonnegative function \(f\) is called a probability density function if \(\int_{-\infty}^{\infty} f(t) d t=1 .\) The probability that \(x\) lies between \(a\) and \(b\) is given by \(P(a \leq x \leq b)=\int_{a}^{b} f(t) d t\) The expected value of \(x\) is given by \(E(x)=\int_{-\infty}^{\infty} t f(t) d t\). Show that the nonnegative function is a probability density function, (b) find \(P(0 \leq x \leq 4),\) and (c) find \(E(x)\). $$ f(t)=\left\\{\begin{array}{ll} \frac{1}{7} e^{-t / 7}, & t \geq 0 \\ 0, & t<0 \end{array}\right. $$
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 \(\frac{f^{\prime}(c)}{g^{\prime}(c)}=\frac{f(b)-f(a)}{g(b)-g(a)}\) \(\begin{array}{l} \underline{\text { Functions }} \\ f(x)=\ln x, \quad g(x)=x^{3} \end{array} \quad \frac{\text { Interval }}{\left[1,4\right]}\)
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. $$ f(t)=\sin a t $$
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