Integration of Piecewise Continuous Functions

Q.

If $\cos \theta =\left(\frac{1}{2}\right)\left(a+\frac{1}{a}\right)$, then the value of $\cos 3\theta$ is

1. $\left(\frac{1}{8}\right)\left(a^3+\frac{1}{a^3}\right)$

2. $\left(\frac{3}{2}\right)\left(a+\frac{1}{a}\right)$

3. $\left(\frac{1}{2}\right)\left(a^3+\frac{1}{a^3}\right)$

4. $\left(\frac{1}{3}\right)\left(a^3+\frac{1}{a^3}\right)$

Q.

Evaluate the following limit:
$\underset{r\to 1}{\text{lim}}\pi r^2$

Q. 19. The minimum value of f (x)=|x-3|+|2+x|+|5-x| is

Q. If x is positive, the sum to infinity of the series
$\frac{1}{1+x}-\frac{1-x}{(1+x{)}^2}+\frac{(1-x{)}^2}{(1+x{)}^3}-\frac{(1-x{)}^3}{(1+x{)}^4}+......\mathrm{is}$
(a) 1/2
(b) 3/4
(c) 1
(d) none of these

Q. The value of $\alpha$ for which $4\alpha \underset{-1}{\overset{2}{\int }}{e}^{-\alpha |x|}dx=5$, is :

1. ${\log }_{e}2$
2. ${\log }_e\sqrt{2}$
3. ${\log }_e\left(\frac{4}{3}\right)$
4. ${\log }_e\left(\frac{3}{2}\right)$

Q. Evaluate following limits
${lim}_{X\to \frac{\pi }{2}}\frac{2^{-\cos x}-1}{x(x-\frac{\pi }{2})}$

Q. The value of integral $\int e^x(\frac{2\tan x}{1+\tan x}+{\cot }^2(x+\frac{\pi }{4}))dx$ is equal to, where $C$ is constant of integration

1. $e^x\tan (x-\frac{\pi }{4})+C$
2. $e^x\tan (\frac{\pi }{4}-x)+C$
3. $e^x\tan (x-\frac{3\pi }{4})+C$
4. $e^x\tan (\frac{3\pi }{4}-x)+C$

Q.

${\int }_0^{\frac{\pi }{2}}\frac{\sqrt{cotx}}{\sqrt{cotx}+\sqrt{tanx}}dx=$ [MP PET 1990, 95; IIT 1983; MNR 1990]

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

Q. If for a real number $y,\left[y\right]$ is the greatest integer less than or equal to $y$, then value of the integral $\underset{\frac{\pi }{2}}{\overset{\frac{3\pi }{2}}{\int }}\left[2\sin x\right]dx$, is

1. $-\pi$
2. $0$
3. $-\frac{\pi }{2}$
4. $\frac{\pi }{2}$

Q. If $f\left(x\right)={\int }_0^{1}f\left(xt\right)dt,$ where $f\left(x\right)$ is a continuos function such that $f\left(1\right)=2$, then

1. $f\left(x\right)$ is a periodic function
2. $f\left(4\right)=2$
3. $f\left(x\right)$ is an even function
4. $f\left(4\right)=5$

Q. 7. | |x-2|-3|>1 then x belong to

Q. If $[.]$ denotes the greatest function, then answer the following by appropriately matching the lists based ​​​​​​on the information given in $\text{Column I}$ and $\text{Column II}$
$\begin{array}{cc}ColumnI& ColumnII\\ \text{a.}\underset{-1}{\overset{1}{\int }}[x+[x+\left[x\right]\left]\right]dx& \text{p.}3\\ \text{b.}\underset{2}{\overset{5}{\int }}\left(\right[x]+[-x\left]\right)dx& \text{q.}5\\ \text{c.}\underset{-1}{\overset{3}{\int }}\text{sgn}(x-[x\left]\right)dx& \text{r.}4\\ \text{d.}25\underset{0}{\overset{\pi /4}{\int }}\left(\right({\tan }^6(x-[x\left]\right)+{\tan }^4(x-[x\left]\right))dx& \text{s.}-3\end{array}$

1. $\left(a\right)\to \left(s\right),\left(b\right)\to \left(q\right),\left(c\right)\to \left(r\right),\left(d\right)\to \left(s\right)$
2. $\left(a\right)\to \left(s\right),\left(b\right)\to \left(r\right),\left(c\right)\to \left(r\right),\left(d\right)\to \left(q\right)$
3. $\left(a\right)\to \left(s\right),\left(b\right)\to \left(s\right),\left(c\right)\to \left(r\right),\left(d\right)\to \left(q\right)$
4. $\left(a\right)\to \left(q\right),\left(b\right)\to \left(s\right),\left(c\right)\to \left(r\right),\left(d\right)\to \left(p\right)$

Q. The value of definite integral ${\int }_{-1}^2\left(x^3-x\right|\mathrm{dx}$ is

1. $2$
2. $3$
3. $\frac{13}{4}$
4. $\frac{11}{4}$

Q.

${\int }_0^2\left(\right[x{]}^2-\left[x^2\right])dx$ is equal to

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

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

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

4. None of these

Q. The value of $\underset{-1}{\overset{3/2}{\int }}|x\sin \pi x|dx=\frac{k\pi +1}{{\pi }^m}$, then $k+m=$

Q. ${\int }_{-\pi }^{\pi }\frac{2x(1+sinx)}{1+co{s}^{2}x}dx=$

1. $\frac{{\pi }^2}{4}$
2. ${\pi }^2$
3. $0$
4. $\frac{\pi }{2}$