Problem 5
Evaluate \(\int_{0}^{1}\left(1+5 x-x^{5}\right)^{4}\left(x^{2}-1\right)\left(x^{2}+1\right) d x\)
Problem 5
Is each of the following integrals defined ? Give a reason for your answer. (a) \(\int_{0}^{1} \frac{\sin x}{x} d x\) (b) \(\int_{0}^{1 / 2} \frac{\tan 2 x}{x} d x\) (c) \(\int_{0}^{1} \frac{1}{x} d x\) (d) \(\int_{0}^{1 / e} \frac{1}{\ln x} d x\) (e) \(\int_{0}^{e} \ln x d x\)
Problem 5
If \(\mathrm{x}=\int_{0}^{y} \frac{\mathrm{dt}}{\sqrt{1+4 \mathrm{t}^{2}}}\) and \(\frac{\mathrm{d}^{2} \mathrm{y}}{\mathrm{dx}^{2}}=\mathrm{ky}\) then find \(\mathrm{k}\)
Problem 5
Evaluate the following integrals : (i) \(\int_{0}^{1}\left(1-x^{2}\right)^{n} d x\) (ii) \(\int_{0}^{1} \frac{x^{2 n} d x}{\sqrt{1-x^{2}}}\) (iii) \(\int_{0}^{2 \mathrm{a}} \mathrm{x}^{9 / 2}(2 \mathrm{a}-\mathrm{x})^{-1 / 2} \mathrm{dx}\) (iv) \(\int_{0}^{\infty} \frac{x^{4} d x}{\left(a^{2}+x^{2}\right)^{2}}\)
Problem 5
Prove that (a) \(\pi=\lim _{n \rightarrow \infty} \frac{4}{n^{2}}\left(\sqrt{n^{2}-1}+\sqrt{n^{2}-2^{2}}+\ldots+\sqrt{n^{2}-n^{2}}\right)\). (b) \(\int_{1}^{3}\left(x^{2}+1\right) d x=\lim _{n \rightarrow \infty} \frac{4}{n^{3}} \sum_{i=1}^{n}\left(n^{2}+2 i n+2 i^{2}\right)\).
Problem 5
Suppose that the function \(\mathrm{f}\) is defined for all \(\mathrm{x}\) such that \(|\mathrm{x}|>1\) and has the property that \(\mathrm{f}^{\prime}(\mathrm{x})=\frac{1}{\mathrm{x} \sqrt{\mathrm{x}^{2}-1}}\) for all such \(\mathrm{x}\). (a) Explain why there exists two constants \(\mathrm{A}\) and \(\mathrm{B}\) such that \(f(x)=\sec ^{-1} x+A\) if \(x>1\) \(f(x)=-\sec ^{-1} x+B\) if \(x<-1\) (b) Determine the values of \(\mathrm{A}\) and \(\mathrm{B}\) so that \(f(2)=1=\mathrm{f}(-2)\). Then sketch the graph of \(y=f(x)\)
Problem 5
Evaluate \(\int_{0}^{a}\left(a^{2}-x^{2}\right)^{5 / 2} d x\)
Problem 5
Find the derivative of \(\mathrm{y}=\int_{1-3 \times}^{1} \frac{\mathrm{u}^{3}}{1+\mathrm{u}^{2}} \mathrm{du}\).
Problem 5
Evaluate the following integrals as limit of sums: (i) \(\int_{1}^{4}\left(x^{2}-x\right) d x\) (ii) \(\int_{0}^{3}\left(x^{2}+1\right) d x\) (iii) \(\int_{0}^{2}\left(2 x^{3}+5\right) d x\) (iv) \(\int_{0}^{1}\left(x+e^{2 x}\right) d x\) (v) \(\int_{0}^{\pi / 2} \cos x d x\) (vi) \(\int_{a}^{b} \sin x d x\)
Problem 5
Prove that \(\lim _{\omega \rightarrow \infty} \frac{e^{k m^{2} x^{2}}}{\int_{a}^{b} e^{k m^{2} x^{2}} d x}= \begin{cases}0 & \text { if } x0, \mathrm{k}>0, \mathrm{~b}>\mathrm{a}>0)\)
