91Ó°ÊÓ

Derive a reduction formula and compute the integral \(\int_{-1}^{0} x^{n} e^{x} d x,(n\) is a positive integer \()\).

A periodic function with period 1 is integrable over any finite interval. Also for two real numbers \(\mathrm{a}, \mathrm{b}\) and for two unequal non-zero postive integers \(\mathrm{m}\) and \(\mathrm{n}, \int_{a}^{a+1} \mathrm{f}(\mathrm{x}) \mathrm{d} \mathrm{x}=\int_{\mathrm{b}}^{b+\mathrm{m}} \mathrm{f}(\mathrm{x}) \mathrm{dx} .\) Calculate the value of \(\int_{\mathrm{m}}^{\mathrm{n}} \mathrm{f}(\mathrm{x}) \mathrm{dx}\)

Evaluate the following integrals: (i) \(\int_{-\infty}^{\infty} \frac{x d x}{x^{4}+1}\) (ii) \(\int_{0}^{1} \frac{\ln (1-x)}{x} \mathrm{dx}\) (iii) \(\int_{0}^{\infty} \frac{\mathrm{dx}}{(\mathrm{x}+1)(\mathrm{x}+2)}\) (iv) \(\int_{0}^{\infty} \frac{x^{2} d x}{\left(x^{2}+a^{2}\right)\left(x^{2}+b^{2}\right)}, a, b>0 .\)

Evaluate the following limits: (i) \(\lim _{n \rightarrow \infty}\left(\frac{n+1}{n^{2}+1^{2}}+\frac{n+2}{n^{2}+2^{2}}+\ldots .+\frac{1}{n}\right)\) (ii) \(\lim _{n \rightarrow \infty} \frac{2^{k}+4^{k}+6^{k}+. .+(2 n)^{k}}{n^{k+1}}, k \neq-1\) (iii) \(\lim _{n \rightarrow \infty} \frac{3}{n}\left[1+\sqrt{\frac{n}{n+3}}+\sqrt{\frac{n}{n+6}}+\sqrt{\frac{n}{n+9}}+\ldots . .\right.\) \(\left.\ldots+\sqrt{\frac{n}{n+3(n-1)}}\right]\) (iv) \(\lim _{n \rightarrow x} \frac{n^{2}}{\left(n^{2}+1\right)^{3 / 2}}+\frac{n^{2}}{\left(n^{2}+2^{2}\right)^{3 / 2}}+\) \(\ldots+\frac{\mathrm{n}^{2}}{\left[\mathrm{n}^{2}+(\mathrm{n}-1)^{2}\right]^{3 / 2}}\)

A function \(\mathrm{f}\) is defined for all real \(\mathrm{x}\) by the formula \(\mathrm{f}(\mathrm{x})=3+\int_{0}^{\mathrm{x}} \frac{1+\sin \mathrm{t}}{2+\mathrm{t}^{2}} \mathrm{dt}\). Without attempting to evaluate this integral, find a quadratic polynomial \(\mathrm{p}(\mathrm{x})=\mathrm{a}+\mathrm{bx}+\mathrm{cx}^{2}\) such that \(\mathrm{p}(0)=\mathrm{f}(0), \mathrm{p}^{\prime}(0)=\mathrm{f}^{\prime}(0)\), and \(\mathrm{p}^{\prime \prime}(0)=\) f' \((0)\).

Determine a region whose area is equal to the limit \(\lim _{n \rightarrow \infty} \sum_{i=1}^{n} \frac{\pi}{4 n} \tan \frac{i \pi}{4 n}\). Donot evaluate the limit.

Assume that \(\mathrm{f}\) is integrable and nonnegative on \([a, b] .\) If \(\int_{a}^{b} f(x) d x=0\), prove that \(f(c)=0\) at each point of continuity of \(f\).

Show that \(0.78<\int_{0}^{1} \frac{d x}{\sqrt{1+x^{4}}}<0.93\)

Prove the inequalities: (i) \(\int_{1}^{3} \sqrt{x^{4}+1} d x \geq \frac{26}{3}\)(iii) \(\frac{1}{17} \leq \int_{1}^{2} \frac{1}{1+x^{4}} \mathrm{dx} \leq \frac{7}{24}\).

Show that the function \(f(x)=\left\\{\begin{array}{cl}\frac{x \ell n x}{1-x}, & 0