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Chapter 6: Problem 2
Decide whether the integral is improper. Explain your reasoning. $$ \int_{1}^{3} \frac{d x}{x^{2}} $$
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Find the integral. Use a computer algebra system to confirm your result. $$ \int \frac{\cot ^{2} t}{\csc t} d t $$
Evaluate \(\lim _{x \rightarrow \infty}\left[\frac{1}{x} \cdot \frac{a^{x}-1}{a-1}\right]^{1 / x}\) where \(a>0, \quad a \neq 1\).
(A) find the indefinite integral in two different ways. (B) Use a graphing utility to graph the antiderivative (without the constant of integration) obtained by each method to show that the results differ only by a constant. (C) Verify analytically that the results differ only by a constant. $$ \int \sec ^{4} 3 x \tan ^{3} 3 x d x $$
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 \(E(x)\).$$ f(t)=\left\\{\begin{array}{ll} \frac{2}{5} e^{-2 t / 5}, & t \geq 0 \\ 0, & t<0 \end{array}\right. $$
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 \(g^{\prime}(0)\).
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