Chapter 6: Problem 1
Decide whether the integral is improper. Explain your reasoning. $$ \int_{0}^{1} \frac{d x}{3 x-2} $$
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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. $$ f(t)=1 $$
Describe the different types of improper integrals
Use a computer algebra system to evaluate the definite integral. In your own words, describe how you would integrate \(\int \sec ^{m} x \tan ^{n} x d x\) for each condition. (a) \(m\) is positive and even. (b) \(n\) is positive and odd. (c) \(n\) is positive and even, and there are no secant factors. (d) \(m\) is positive and odd, and there are no tangent factors.
A "semi-infinite" uniform rod occupies the nonnegative \(x\) -axis. The rod has a linear density \(\delta\) which means that a segment of length \(d x\) has a mass of \(\delta d x .\) A particle of mass \(m\) is located at the point \((-a, 0)\). The gravitational force \(F\) that the rod exerts on the mass is given by \(F=\int_{0}^{\infty} \frac{G M \delta}{(a+x)^{2}} d x\) where \(G\) is the gravitational constant. Find \(F\).
Prove that if \(f(x) \geq 0, \lim _{x \rightarrow a} f(x)=0,\) and \(\lim _{x \rightarrow a} g(x)=-\infty,\) then \(\lim _{x \rightarrow a} f(x)^{g(x)}=\infty\)
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