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Chapter 4: Problem 3
Find the indefinite integral. $$ \int \frac{1}{3-2 x} d x $$
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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 \(1 \leq t \leq 4\).
Find \(F^{\prime}(x)\). $$ F(x)=\int_{0}^{x^{3}} \sin t^{2} d t $$
In Exercises \(79-84,\) find \(F^{\prime}(x)\). $$ F(x)=\int_{x}^{x+2}(4 t+1) d t $$
(a) integrate to find \(F\) as a function of \(x\) and (b) demonstrate the Second Fundamental Theorem of Calculus by differentiating the result in part (a). $$ F(x)=\int_{-1}^{x} e^{t} d t $$
In Exercises \(88-92,\) verify the differentiation formula. \(\frac{d}{d x}[\cosh x]=\sinh x\)
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