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Chapter 4: Problem 21
Find the indefinite integral. $$ \int \frac{\cos t}{1+\sin t} d t $$
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Find the integral. \(\int x \operatorname{csch}^{2} \frac{x^{2}}{2} d x\)
In Exercises \(47-52,\) evaluate the integral. \(\int_{0}^{\ln 2} \tanh x d x\)
A horizontal plane is ruled with parallel lines 2 inches apart. A two-inch needle is tossed randomly onto the plane. The probability that the needle will touch a line is \(P=\frac{2}{\pi} \int_{0}^{\pi / 2} \sin \theta d \theta\) where \(\theta\) is the acute angle between the needle and any one of the parallel lines. Find this probability.
Verify the differentiation formula. \(\frac{d}{d x}\left[\sinh ^{-1} x\right]=\frac{1}{\sqrt{x^{2}+1}}\)
From the vertex \((0, c)\) of the catenary \(y=c \cosh (x / c)\) a line \(L\) is drawn perpendicular to the tangent to the catenary at a point \(P\). Prove that the length of \(L\) intercepted by the axes is equal to the ordinate \(y\) of the point \(P\).
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