Integration by Substitution

Q.

Find the derivative of $secx$.

Q.

$\frac{d}{dx}\left(x^2{e}^x\sin \left(x\right)\right)=$

1. $x{e}^x\left(2\sin \left(x\right)+x\sin \left(x\right)+x\cos \left(x\right)\right)$

2. $x{e}^x\left(2\sin \left(x\right)+x\sin \left(x\right)-\cos \left(x\right)\right)$

3. $x{e}^x\left(2\sin \left(x\right)+x\sin \left(x\right)+\cos \left(x\right)\right)$

4. none of these

Q.

Prove:$\frac{1+{\tan }^{2}A}{1+co{t}^{2}A}={\left[\frac{1-\tan A}{1-cotA}\right]}^2={\tan }^{2}A.$

Q.

What is the derivative of ${\cot }^{2}x$ ?

Q.

If $\begin{array}{rcl}\frac{{\left(1+i\right)}^2}{\left(2-i\right)}& =& x+iy\end{array}$, then $x+y$ is

1. $\frac{-2}{5}$

2. $\frac{6}{5}$

3. $\frac{2}{5}$

4. $\frac{-6}{5}$

Q.

If $y=\frac{e^x\log x}{x^2}$, then $dy/dx$ is equal to ?

1. $e^x[1+(x+2)\log x]/x^4$

2. $e^x[1+(x-2)\log x]/x^4$

3. $e^x[1–(x–2)\log x]/x^3$

4. $e^x[1+(x–2)\log x]/x^3$

Q.

The total number of positive integral solutions $(x,y,z)$ such that $xyz=24$ is

1. $36$

2. $45$

3. $24$

4. $30$

Q.

If $co{t}^{-1}\left(x\right)+{\tan }^{-1}\left(3\right)=\frac{\pi }{2}$, then $x=$

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

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

3. $3$

4. $4$

Q.

The maximum value of $f\left(x\right)=\sin x(1+\cos x)$ is

1. $\raisebox{1ex}{$3\sqrt{3}$}\!\left/ \!\raisebox{-1ex}{$4$}\right.$

2. $\raisebox{1ex}{$3\sqrt{3}$}\!\left/ \!\raisebox{-1ex}{$2$}\right.$

3. $3\sqrt{3}$

4. $\sqrt{3}$

Q.

How do you simplify $\frac{1-{\tan }^{2}x}{1+{\tan }^{2}x}$?

Q.

The solution set of the equation $(x+1)(x+2)(x+3)(x+4)=120$ is

1. $\left\{-6,1,\frac{\left[-5\pm \sqrt{39}i\right]}{2}\right\}$

2. $\left\{6,-1,\frac{\left[-5\pm \sqrt{39i}\right]}{2}\right\}$

3. $\left\{-6,-1,\frac{\left[-5\pm \sqrt{39i}\right]}{2}\right\}$

4. $\left\{6,1,\frac{\left[-5\pm \sqrt{39i}\right]}{2}\right\}$

Q.

The integral ${\int }_0^2\left|\left|x-1\right|-x\right|dx$ is equal to :

Q.

If $x=a\sin \theta$ and $y=b\cos \theta$, then $d^{2}y/d{x}^2$ is:

1. $(a/b^2)se{c}^2\theta$

2. $-(b/a)se{c}^2\theta$

3. $-(b/a^2)se{c}^3\theta$

4. $(b^2/a^2)se{c}^3\theta$

Q.

Let $\mathrm{f}(\mathrm{x}+\mathrm{y})=\mathrm{f}\left(\mathrm{x}\right)\mathrm{f}\left(\mathrm{y}\right)$ for all $\mathrm{x}$ and $\mathrm{y}$, suppose $\mathrm{f}\left(5\right)=2$, and$\mathrm{f}’\left(0\right)=3$, then $\mathrm{f}’\left(5\right)$ equals to

1. $6$

2. $7$

3. $4$

4. $8$

Q. The inegral $\int \frac{3x^{13}+2x^{11}}{(2x^4+3x^2+1{)}^4}dx$ is equal to :
(where $C$ is a constant of integration)

1. $\frac{x^4}{6(2x^4+3x^2+1{)}^3}+C$
2. $\frac{x^{12}}{6(2x^4+3x^2+1{)}^3}+C$
3. $\frac{x^{12}}{(2x^4+3x^2+1{)}^3}+C$
4. $\frac{x^4}{(2x^4+3x^2+1{)}^3}+C$

Q. The integral $\int \frac{dx}{(1+\sqrt{x})\sqrt{x-x^2}}$ is equal to: (where $C$ is a constant of integration).

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

Q.

$\int e^{-\log x}dx=$

1. $e^{-\log x}+c$

2. $x{e}^{-\log x}+c$

3. $e^{\log x}+c$

4. $\log \left|x\right|+c$

Q.

If $\mathrm{y}={\mathrm{x}}^{\mathrm{x}}$, then $\frac{\mathrm{dy}}{\mathrm{dx}}$ is

1. ${\mathrm{x}}^{\mathrm{x}}\log {\mathrm{e}}^{\mathrm{x}}$

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

3. ${\mathrm{x}}^{\mathrm{x}}(1+\log \mathrm{x})$

4. ${\mathrm{x}}^{\mathrm{x}}\log \mathrm{x}$

Q.

If $\tan \left(A\right)=\frac{1}{2},\tan \left(B\right)=\frac{1}{3}$, then $\cos \left(2A\right)=$

1. $\sin \left(B\right)$

2. $\sin \left(2B\right)$

3. $\sin \left(3B\right)$

4. None of these

Q.

If $y=\frac{e^{2x}+e^{-2x}}{e^{2x}-e^{-2x}}$, then $\frac{dy}{dx}=?$

1. $\frac{-8}{{\left(e^{2x}-e^{-2x}\right)}^2}$

2. $\frac{8}{{\left(e^{2x}-e^{-2x}\right)}^2}$

3. $\frac{-4}{{\left(e^{2x}-e^{-2x}\right)}^2}$

4. $\frac{4}{{\left(e^{2x}-e^{-2x}\right)}^2}$

Q.

$(d/dx)[\cos {(1-x2)}^2]=$

1. $-2x(1-x^2)\sin {(1-x^2)}^2$

2. $-4x(1-x^2)\sin {(1-x^2)}^2$

3. $4x(1-x^2)\sin {(1-x^2)}^2$

4. $noneofthese$

Q. If $\sin x+\cos x=\sqrt{2}\cos x$, then cosx - sin x is equal to

1. 
2. 
3. 
4. 

Q.

What is the formula for $\tan 3x$?

Q. If $\int \frac{dx}{(x^2-2x+10{)}^2}=A({\tan }^{-1}\left(\frac{x-1}{3}\right)+\frac{f\left(x\right)}{x^2-2x+10})+C$ , where $C$ is a constant of integeration, then :

1. $A=\frac{1}{81}$ and $f\left(x\right)=3(x-1)$
2. $A=\frac{1}{27}$ and $f\left(x\right)=9(x-1)$
3. $A=\frac{1}{54}$ and $f\left(x\right)=3(x-1)$
4. $A=\frac{1}{54}$ and $f\left(x\right)=9(x-1{)}^2$

Q.

What is$\sin \left(x\right)$ times $\sin \left(x\right)$?

Q.

An object is displaced from position $\overrightarrow{r_1}=\left(2\hat{i}+3\hat{j}\right)m$ to $\overrightarrow{r_2}=\left(4\hat{i}+6\hat{j}\right)m$ under a force $\overrightarrow{F}=\left(3x^2\hat{i}+2y\hat{j}\right)N$ then work done by the force is

1. $24J$

2. $33J$

3. $83J$

4. $45J$

Q.

If $\mathit{a}\mathbf{,}\mathbf{}\mathit{b}and\mathit{c}$ are unit vectors such that$\mathit{a}+\mathit{b}+\mathit{c}=0$, find the value of vector $\mathit{a}\mathbf{.}\mathit{b}+\mathit{b}\mathbf{.}\mathit{c}+\mathit{c}\mathbf{.}\mathit{a}?$

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

2. $0$

3. $1$

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

Q.

Integrate the function.
$\int x^2{e}^{x}dx.$

Q.

If $f:\left[-6,6\right]\to R$ is defined by $f\left(x\right)=x^2-3$ for $x\in R$, then $\left(fofof\right)\left(-1\right)+\left(fofof\right)\left(0\right)+\left(fofof\right)\left(1\right)$ is equal to

1. $f\left(4\sqrt{2}\right)$

2. $f\left(3\sqrt{2}\right)$

3. $f\left(2\sqrt{2}\right)$

4. $f\left(\sqrt{2}\right)$

Q. Let $\int e^{x^2}\cdot e^x(2x^2+x+1)dx=e^{x^2}f\left(x\right)+c$, where $c$ is constant of integration. If the minimum value of $f\left(x\right)$ is $m$, then the value of $[-\frac{1}{m}]$ is
(where [.] represents grestest integer function)