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Chapter 4: Problem 6
Find the indefinite integral. $$ \int \frac{x}{\sqrt{9-x^{2}}} d x $$
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Consider a particle moving along the \(x\) -axis where \(x(t)\) is the position of the particle at time \(t, x^{\prime}(t)\) is its velocity, and \(\int_{a}^{b}\left|x^{\prime}(t)\right| d t\) is the distance the particle travels in the interval of time. A particle moves along the \(x\) -axis with velocity \(v(t)=1 / \sqrt{t}\) \(t > 0\). At time \(t=1,\) its position is \(x=4\). Find the total distance traveled by the particle on the interval \(1 \leq t \leq 4\).
Use the Second Fundamental Theorem of Calculus to find \(F^{\prime}(x)\). $$ F(x)=\int_{1}^{x} \frac{t^{2}}{t^{2}+1} d t $$
Think About It Determine whether the Dirichlet function $$f(x)=\left\\{\begin{array}{ll}1, & x \text { is rational } \\ 0, & x \text { is irrational }\end{array}\right.$$ is integrable on the interval [0,1] . Explain.
Determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false. $$ \int \frac{d x}{25+x^{2}}=\frac{1}{25} \arctan \frac{x}{25}+C $$
Consider the function \(F(x)=\frac{1}{2} \int_{x}^{x+2} \frac{2}{t^{2}+1} d t\) (a) Write a short paragraph giving a geometric interpretation of the function \(F(x)\) relative to the function \(f(x)=\frac{2}{x^{2}+1}\) Use what you have written to guess the value of \(x\) that will make \(F\) maximum. (b) Perform the specified integration to find an alternative form of \(F(x)\). Use calculus to locate the value of \(x\) that will make \(F\) maximum and compare the result with your guess in part (a).
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