Definite Integral as Limit of Sum

Q.

$li{m}_{\pi \to \infty }{\sum }_{k=1}^n\frac{k}{n^2+k^2}$ is equals to [Roorkee 1999]

1. $\frac{1}{2}log2$
2. $log2$
3. $\frac{\pi }{4}$
4. $\frac{\pi }{2}$

Q. If $I=\underset{8}{\overset{16}{\int }}(\sqrt{x+\sqrt{32x-256}}+\sqrt{x-\sqrt{32x-256}})dx$ then ${\left(\frac{I}{16}\right)}^2$ is

Q. The value of $\underset{n\to \infty }{lim}\frac{1}{n^3}(\sqrt{n^2+1}+2\sqrt{n^2+2^2}+\cdots +n\sqrt{n^2+n^2})$ is

1. $\frac{\sqrt{2}-1}{5}$
2. $\frac{\sqrt{2}-1}{3}$
3. $\frac{2\sqrt{2}-1}{5}$
4. $\frac{2\sqrt{2}-1}{3}$

Q.

$={lim}_{n\to \infty }[\frac{1}{n}+\frac{1}{\sqrt{n^2+n}}+\frac{1}{\sqrt{n^2+2n}}+\cdots +\frac{1}{\sqrt{n^2+(n-1)n}}]$ is equal to [RPET 2000]

1. $2+2\sqrt{2}$
2. $2\sqrt{2}-2$
3. $2\sqrt{2}$
4. $2$

Q. ${lim}_{n\to \infty }\frac{1}{1^3+n^3}+\frac{4}{2^3+n^3}+\cdots +\frac{1}{2n}$ is equal to

1. $\frac{1}{3}lo{g}_{e}3$
2. $\frac{1}{3}lo{g}_{e}2$
3. $\frac{1}{3}lo{g}_e\frac{1}{3}$

4. $Noneofthese$

Q. $\text{Let}\int \frac{(x^2-1)dx}{x^3\sqrt{3x^4+2x^2-1}}=f\left(x\right)+c\text{and}\lambda =\underset{x\to \infty }{lim}f\left(x\right)$, then the absolute value of $\frac{\lambda }{\sqrt{3}}$ is (where $c$ is the constant of integration)

Q. The value of $\underset{n\to \infty }{lim}\frac{\pi }{6n}[{\sec }^2\left(\frac{\pi }{6n}\right)+{\sec }^2(2\cdot \frac{\pi }{6n})+\cdots +{\sec }^2\left(\right(n-1\left)\frac{\pi }{6n}\right)+\frac{4}{3}]$ is

1. $\frac{\sqrt{3}}{3}$
2. $\sqrt{3}$
3. $2$
4. $\frac{2}{\sqrt{3}}$

Q. $\underset{n\to \infty }{lim}[\frac{1}{n}+\frac{n}{(n+1{)}^2}+\frac{n}{(n+2{)}^2}+....+\frac{n}{(2n-1{)}^2}]$ is equal to:

1. $1$
2. $\frac{1}{3}$
3. $\frac{1}{2}$
4. $\frac{1}{4}$

Q. ${lim}_{n\to \infty }{\sum }_{r=1}^n\frac{1}{n}{e}^{\frac{r}{n}}$ is

1. e
2. e - 1
3. 1 - e
4. 1 + e

Q. The value of $\underset{n\to \infty }{lim}\frac{1}{n^2}\{{\sin }^2\frac{\pi }{4n}+2{\sin }^2\frac{2\pi }{4n}+\cdots +n{\sin }^2\frac{4\pi }{4n}\}$ is

1. $\frac{1}{2}-\frac{2}{\pi }+\frac{4}{{\pi }^2}$
2. $\frac{1}{4}-\frac{1}{\pi }+\frac{2}{{\pi }^2}$
3. $\frac{1}{4}+\frac{1}{\pi }-\frac{2}{{\pi }^2}$
4. $\frac{1}{2}-\frac{2}{\pi }$

Q.

$li{m}_{\pi \to \infty }\frac{1}{n}{\sum }_{r=1}^{2n}\frac{r}{\sqrt{n^2+r^2}}equals$ [IIT 1997 Re-exam]

1. $1+\sqrt{5}$
2. $-1+\sqrt{5}$
3. $-1+\sqrt{2}$
4. $1+\sqrt{2}$

Q.

${lim}_{n\to \infty }[\frac{1}{n}+\frac{1}{n+1}+\frac{1}{n+2}+\cdots +\frac{1}{2n}]=$ [Karnataka CET 1999]

1. $0$
2. $lo{g}_{e}4$

3. $lo{g}_{e}3$

4. $lo{g}_{e}2$

Q. $\stackrel{lim}{n\to \infty }[\frac{1}{n+1}+\frac{1}{n+2}+\cdots \frac{1}{n+n}]$ is equal to

1. 3 log 2
2. log 2
3. 2 log 2
4. None of these