Problem 5
Evaluate \(\int_{0}^{\pi / 2} \ln (1+\cos \theta \cos x) \frac{d x}{\cos x}\)
Problem 5
Prove that \(\lim _{\omega \rightarrow \infty} \frac{e^{k m^{2} x^{2}}}{\int_{a}^{b} e^{k m^{2} x^{2}} d x}= \begin{cases}0 & \text { if } x0, \mathrm{k}>0, \mathrm{~b}>\mathrm{a}>0)\)
Problem 6
Prove that if \(|x|<1\) \(\frac{x^{3}}{1.3}-\frac{x^{5}}{3.5}+\frac{x^{7}}{5.7}-\ldots=\frac{1}{2}\left(1+x^{2}\right) \tan ^{-1} x-\frac{1}{2} x\)
Problem 6
Estimate \(\int_{0}^{3} \mathrm{f}(\mathrm{x}) \mathrm{d} x\) if \(\mathrm{it}\) is known that \(f(0)=10, f(0.5)=13, f(1)=14, f(1.5)=16, f(2)=18\) \(\mathrm{f}(2.5)=10, \mathrm{f}(3)=6 \mathrm{by}\) (a) the trapezoidal method. (b) Simpson's method.
Problem 6
Replace the symbol \(*\) by either \(\leq\) or \(\geq\) so that the resulting expressions are correct. Give your reasons. (a) \(\int_{0}^{1} x^{2} d x * \int_{0}^{1} x^{3} d x\) (b) \(\int_{-1}^{1} x^{2} d x * \int_{-1}^{1} x^{3} d x\) (c) \(\int_{1}^{3} x^{2} d x * \int_{1}^{3} x^{3} d x\).
Problem 6
If f is continuous and \(\int_{0} \mathrm{f}(\mathrm{x}) \mathrm{d} \mathrm{x}=4\), find \(\int_{0}^{3} x f\left(x^{2}\right) d x\)
Problem 6
Evaluate the following limits: (i) \(\lim _{n \rightarrow x} \frac{1}{n}+\frac{1}{n+1}+\frac{1}{n+2}+\ldots .+\frac{1}{4 n}\) (ii) \(\lim _{n \rightarrow \infty}\left[\frac{1}{n}+\frac{n^{2}}{(n+1)^{3}}+\frac{n^{2}}{(n+2)^{3}} \ldots .+\frac{1}{8 n}\right]\) (iii) \(\lim _{n \rightarrow \infty}\left[\frac{n+1}{n^{2}+1^{2}}+\frac{n+2}{n^{2}+2^{2}}+\frac{n+3}{n^{2}+3^{2}}+\ldots . .+\frac{3}{5 n}\right]\)
Problem 6
Find a function \(\mathrm{f}\) and a value of the constant \(\mathrm{c}\) such that \(\int_{c}^{x} \operatorname{tf}(t) d t=\sin x-x \cos x-\frac{1}{2} x^{2}\) for all real \(\mathrm{x}\).
Problem 6
State whether or not each of the following functions is integrable in the given interval. Give reasons for each answer. (i) \(\mathrm{f}(\mathrm{x})=|\mathrm{x}-1|,[0,3]\). (ii) \(F(x)=\left\\{\begin{array}{cl}1 & \text { if } x \text { rational } \\\ -1 & \text { if x irrational, }[0,1] .\end{array}\right.\)
Problem 6
Let \(\mathrm{f}(\mathrm{x})\) denote a linear function that is nonnegative on the interval \([a, b]\). For each value of \(\mathrm{x}\) in \([\mathrm{a}, \mathrm{b}]\), define \(\mathrm{A}(\mathrm{x})\) to be the area between the graph of \(\mathrm{f}\) and the interval \([\mathrm{a}, \mathrm{x}]\). (a) Prove that \(\mathrm{A}(\mathrm{x})=\frac{1}{2}[\mathrm{f}(\mathrm{a})+\mathrm{f}(\mathrm{x})](\mathrm{x}-\mathrm{a})\). (b) Use part (a) to verify that \(\mathrm{A}^{\prime}(\mathrm{x})=\mathrm{f}(\mathrm{x})\).
