Chapter 2: Problem 5
Evaluate \(\int_{0}^{1}\left(1+5 x-x^{5}\right)^{4}\left(x^{2}-1\right)\left(x^{2}+1\right) d x\)
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\(\sqrt{1}+x\) Prove that, if \(\mathrm{n}>1\) (i) \(0<\int_{0}^{\pi / 2} \sin ^{n+1} x d x<\int_{0}^{\pi / 2} \sin ^{n} x d x\), (ii) \(0<\int_{0}^{\pi / 4} \tan ^{n+1} x d x<\int_{0}^{\pi / 4} \tan ^{n} x d x\). (iii) \(0.5<\int_{0}^{1 / 2} \frac{\mathrm{dx}}{\sqrt{\left(1-\mathrm{x}^{2 \mathrm{a}}\right)}}<0.524\).
Explain why each of the following integrals is improper. (a) \(\int_{1}^{\infty} x^{4} e^{-x^{4}} d x\) (b) \(\int_{0}^{\pi / 2} \sec x d x\) (c) \(\int_{0}^{2} \frac{x}{x^{2}-5 x+6} d x\) (d) \(\int_{-\infty}^{0} \frac{1}{x^{2}+5} d x\)
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) \(\lim _{n \rightarrow \infty}\left[\frac{n+1}{n^{2}+1^{2}}+\frac{n+2}{n^{2}+2^{2}}+\frac{n+3}{n^{2}+3^{2}}+\ldots . .+\frac{3}{5 n}\right]\)
Prove the inequalities: (i) \(\int_{1}^{3} \sqrt{x^{4}+1} d x \geq \frac{26}{3}\)(iii) \(\frac{1}{17} \leq \int_{1}^{2} \frac{1}{1+x^{4}} \mathrm{dx} \leq \frac{7}{24}\).
Given that \(\int_{0}^{1} \frac{\ln x}{(1+x) \sqrt{x}} d x\) is a convergent improper integral, prove that \(\int_{0}^{\infty} \frac{\ln x d x}{(1+x) \sqrt{x}}=0\).
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