Integration of Piecewise Continuous Functions

Q.

How do you integrate $\int \tan xdx$.

Q.

$\frac{d\left[\log \right(\tan x\left)\right]}{dx}=$

1. $2sec2x$

2. $2\cos ec2x$

3. $sec2x$

4. $\cos ec2x$

Q. The value of the integral, $\underset{1}{\overset{3}{\int }}[x^2-2x-2]dx,$ where $\left[x\right]$ denotes the greatest integer less than or equal to $x,$ is

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

Q. If the function $f\left(x\right)=\{\begin{array}{cc}\frac{1}{x}{\log }_e\left(\frac{1+\frac{x}{a}}{1-\frac{x}{b}}\right),& x<0\\ k,& x=0\\ \frac{{\cos }^{2}x-{\sin }^{2}x-1}{\sqrt{x^2+1}-1},& x>0\end{array}$
is continuous at $x=0,$ then $\frac{1}{a}+\frac{1}{b}+\frac{4}{k}$ is equal to:

1. $5$
2. $4$
3. $-4$
4. $-5$

Q.

What is the formula of ${\sin }^3\left(x\right)$​?

Q. If $\frac{dy}{dx}=\frac{2^{x}y+2^y\cdot 2^x}{2^x+2^{x+y}{\log }_e{}^2},$ $y\left(0\right)=0,$ then for $y=1$, the value of $x$ lies in the interval

1. $(\frac{1}{2},1]$
2. $(0,\frac{1}{2}]$
3. $(1,2)$
4. $(2,3)$

Q. Which of the following function(s) not defined at $x=0$ has/have irremovable discontinuity at $x=0$?

1. $f\left(x\right)=\frac{1}{\ln |x|}$
2. $f\left(x\right)=\cos \left(\frac{|\sin x|}{x}\right)$
3. $f\left(x\right)=x\sin \left(\frac{\pi }{x}\right)$
4. $f\left(x\right)=\frac{1}{1+2^{\cot x}}$

Q.

Assertion: The sum of two vectors can be zero. Reason: The vector cancels each other when they are equal and opposite.

1. both Assertion and Reason are true and the Reason is the correct explanation of the Assertion.

2. both Assertion and Reason are true but Reason is not the correct explanation of the Assertion.

3. Assertion is true but Reason is false.

4. Assertion and Reason both are false.

5. Assertion is false but Reason is true

Q.

Differential coefficient of ${\sin }^{-1}\left(x\right)$ w.r.t.${\cos }^{-1}\left(\sqrt{\left(1-x^2\right)}\right)$ is

1. $1$

2. $2$

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

4. none of these

Q. Let $f:R\to R$ be a function. We say that $f$ has
PROPERTY $1$ if $\underset{h\to 0}{lim}\frac{f\left(h\right)-f\left(0\right)}{\sqrt{|h|}}$ exists and is finite, and
PROPERTY $2$ if $\underset{h\to 0}{lim}\frac{f\left(h\right)-f\left(0\right)}{h^2}$exists and is finite.
Then which of the following options is/are correct?

1. $f\left(x\right)=|x|$ has PROPERTY $1$
2. $f\left(x\right)=x^{\frac{2}{3}}$ has PROPERTY $1$
3. $f\left(x\right)=x|x|$ has PROPERTY $2$
4. $f\left(x\right)=\sin x$ has PROPERTY $2$

Q. $f\left(x\right)$ is a polynomial such that $f\left(x\right)\cdot f\left(\frac{1}{x}\right)=f\left(x\right)+f\left(\frac{1}{x}\right)$ such that $f\left(3\right)=28$. Then the value of $\sum _{k=1}^{10}f\left(k\right)$ is

Q.

Integrate the following functions.
$\int \frac{1}{9x^2+6x+5}dx.$

Q. The value of $\sum _{n=1}^{10}\underset{-2n-1}{\overset{-2n}{\int }}{\sin }^{27}xdx+\sum _{n=1}^{10}\underset{2n}{\overset{2n+1}{\int }}{\sin }^{27}xdx=$

1. $27$
2. $54$
3. $-54$
4. $0$

Q.

Evaluate :${\int }_0^{\frac{\mathrm{\pi }}{2}}\frac{xSinx\cos x}{{\cos }^{4}x+{\sin }^{4}x}dx$

1. $\frac{{\mathrm{\pi }}^2}{4}$

2. $\frac{{\mathrm{\pi }}^2}{8}$

3. $\frac{{\mathrm{\pi }}^2}{16}$

4. $\frac{{\mathrm{\pi }}^2}{32}$

Q. If x= 7+40, find the value of x+ 1/x

Q.

${\int }_0^{\frac{\mathrm{\pi }}{2}}\frac{\sqrt{cotx}}{\sqrt{cotx}+\sqrt{\tan x}}dx=$

1. $1$

2. $-1$

3. $\frac{\mathrm{\pi }}{2}$

4. $\frac{\mathrm{\pi }}{4}$

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. Let $f\left(x\right)$ be a differentiable function such that $f^{\prime }\left(0\right)=1,$ and the sequence $\left\{a_n\right\}$ is defined as $a_1=2$ and $a_n=\underset{x\to \infty }{lim}{x}^2{(f\left(\frac{a_{n-1}}{x}\right)-f(0\left)\right)}^2$ for $n\ge 2.$ If the value of $\prod _{i=1}^{10}{a}_i=2^k,$ then the value of $k$ is

Q.

Evaluate : ${\int }_0^{\mathrm{\pi }}\sqrt{1+4{\sin }^2\left(x/2\right)-4\sin \left(x/2\right)}dx$

1. $\mathrm{\pi }-4$

2. $\left(2\mathrm{\pi }/3\right)-4-4\sqrt{3}$

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

4. $4\sqrt{3}-4-\left(\mathrm{\pi }/3\right)$

Q.

The value of $I={\int }_0^3\left(\right[x]+[x+\frac{1}{3}]+[x+\frac{2}{3}])dx$, where $[\cdot ]$ denotes the greatest integer function, is equal to

1. 10
   

2. 11
   

3. 12
   

4. 14

Q. $f$ is an odd function. It is also known that $f\left(x\right)$ is continuous for all values of $x$ and is periodic with period $2.$ If $g\left(x\right)=\underset{0}{\overset{x}{\int }}f\left(t\right)dt,$ then

1. $g\left(x\right)$ is even
2. $g\left(n\right)=0,n\in N$
3. $g\left(2n\right)=0,n\in N$
4. $g\left(x\right)$ is non-periodic

Q.

The second-order derivative of $a{\sin }^3\left(t\right)$ with respect to $a{\cos }^3\left(t\right)$ at $t=\frac{\mathrm{\pi }}{4}$ is

1. $2$

2. $\frac{1}{12a}$

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

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

Q. If $C_1,C_2$ are the values of $x,(C_1<C_2)$ for which LMVT holds for the function $f\left(x\right)=x^3$ on the interval $[-3,3],$ then the value of $4C_1^2+7C_2^2$ is equal to

Q. If $f\left(x\right)=\frac{x}{1+\left({\log }_{e}x\right)\left({\log }_{e}x\right)\cdots \infty }\forall x\in [1,3]$ is non-differentiable at $x=k,$ then the value of $\left[k^2\right],$ is (where $[.]$ denotes the greatest integer function)

Q. If ${\int }_{-1}^{4}f\left(x\right)dx=4$ and ${\int }_2^4[3-f(x\left)\right]dx=7then{\int }_{-1}^{2}f\left(x\right)dx=$

1. 5
2. 8
3. -2
4. 3

Q. If $f\left(x\right)=\frac{(-e^x+2^x)}{x}$ is continuous at $x=0$, then the value of $10|f\left(0\right)-{\log }_{e}2|$ is

Q.

If ${\int }_0^{25}{e}^{x-\left[x\right]}dx=k(e-1)$, then the value of $k$ is

1. $12$

2. $25$

3. $23$

4. $24$

Q. ${\int }_{-1}^1\frac{3x^{2}dx}{1+4^{tanx}}=$

1. 0
2. -1
3. 1
4. 2

Q. The value of $[{\int }_0^{\frac{\pi }{4}}\frac{2(1+x)\ln (1+x)\cdot \sin x+\cos x}{2(1+x){\cos }^{3}x}dx]$ is (where [.] denotes the greatest integer function)

Q. If $I_n=\underset{\frac{\pi }{4}}{\overset{\frac{\pi }{2}}{\int }}{\cot }^{n}xdx$, then

1. $\frac{1}{I_2+I_4},\frac{1}{I_3+I_5},\frac{1}{I_4+I_6}\text{are in G.P.}$
2. $\frac{1}{I_2+I_4},\frac{1}{I_3+I_5},\frac{1}{I_4+I_6}\text{are in A.P.}$
3. $I_2+I_4,I_3+I_5,I_4+I_6\text{are in A.P.}$
4. $I_2+I_4,(I_3+I_5{)}^2,I_4+I_6\text{are in G.P.}$