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Chapter 6: Problem 24
Use integration tables to evaluate the integral. $$ \int_{0}^{\pi} x \sin x d x $$
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For what value of \(c\) does the integral \(\int_{0}^{\infty}\left(\frac{1}{\sqrt{x^{2}+1}}-\frac{c}{x+1}\right) d x\) converge? Evaluate the integral for this value of \(c\).
Find the integral. Use a computer algebra system to confirm your result. $$ \int \frac{1-\sec t}{\cos t-1} d t $$
Find the area of the region bounded by the graphs of the equations.$$ y=\sin x, \quad y=\sin ^{3} x, \quad x=0, \quad x=\pi / 2 $$
(a) Let \(f^{\prime}(x)\) be continuous. Show that \(\lim _{h \rightarrow 0} \frac{f(x+h)-f(x-h)}{2 h}=f^{\prime}(x)\) (b) Explain the result of part (a) graphically.
Prove that \(I_{n}=\left(\frac{n-1}{n+2}\right) I_{n-1},\) where \(I_{n}=\int_{0}^{\infty} \frac{x^{2 n-1}}{\left(x^{2}+1\right)^{n+3}} d x, \quad n \geq 1 .\) Then evaluate each integral. (a) \(\int_{0}^{\infty} \frac{x}{\left(x^{2}+1\right)^{4}} d x\) (b) \(\int_{0}^{\infty} \frac{x^{3}}{\left(x^{2}+1\right)^{5}} d x\) (c) \(\int_{0}^{\infty} \frac{x^{5}}{\left(x^{2}+1\right)^{6}} d x\)
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