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Chapter 6: Problem 26
Use integration tables to evaluate the integral. $$ \int_{2}^{4} \frac{x^{2}}{(3 x-5)^{2}} d x $$
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Sketch the graph of \(g(x)=\left\\{\begin{array}{ll}e^{-1 / x^{2}}, & x \neq 0 \\ 0, & x=0\end{array}\right.\) and determine \(g^{\prime}(0)\).
Explain why \(\int_{-1}^{1} \frac{1}{x^{3}} d x \neq 0\)
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 \(E(x)\). $$ f(t)=\left\\{\begin{array}{ll} \frac{1}{7} e^{-t / 7}, & t \geq 0 \\ 0, & t<0 \end{array}\right. $$
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. $$ f(t)=\cos a t $$
Define the terms converges and diverges when working with improper integrals.
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