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Chapter 4: Problem 42
Use a computer algebra system to find or evaluate the integral. $$ \int \frac{x^{2}}{x-1} d x $$
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Prove or disprove that there is at least one straight line normal to the graph of \(y=\cosh x\) at a point \((a, \cosh a)\) and also normal to the graph of \(y=\sinh x\) at a point \((c, \sinh c)\). [At a point on a graph, the normal line is the perpendicular to the tangent at that point. Also, \(\cosh x=\left(e^{x}+e^{-x}\right) / 2\) and \(\left.\sinh x=\left(e^{x}-e^{-x}\right) / 2 .\right]\)
Determine \(\lim _{n \rightarrow \infty} \frac{1}{n^{3}}\left[1^{2}+2^{2}+3^{2}+\cdots+n^{2}\right]\) by using an appropriate Riemann sum.
Verify the differentiation formula. \(\frac{d}{d x}\left[\cosh ^{-1} x\right]=\frac{1}{\sqrt{x^{2}-1}}\)
Find all the continuous positive functions \(f(x),\) for \(0 \leq x \leq\) such that \(\int_{0}^{1} f(x) d x=1, \int_{0}^{1} f(x) x d x=\alpha,\) and \(\int_{0}^{1} f(x) x^{2} d x=\alpha^{2}\) where \(\alpha\) is a real number
Evaluate the integral. \(\int_{0}^{4} \frac{1}{25-x^{2}} d x\)
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