Integration by Parts

Q.

$\int \left(\frac{1}{x}\right)\log xdx=$

1. ${\log }_e\left(1-{\log }_{e}x\right)+C$

2. ${\log }_e\left({\log }_{e}ex-1\right)+C$

3. $\log \left(\log x-1\right)+C$

4. ${\log }_e\left({\log }_{e}x+x\right)+C$

5. $\log \left(\log x\right)+C$

Q.

Integrate $\int \frac{1}{1+\sin x}dx$ ?

Q.

The coefficient of $x^4$ in the expansion of ${(1+x+x^2+x^3)}^6$ in powers of $x$, is

1. $110$

2. $120$

3. $130$

4. $100$

Q.

$\frac{\tan \left(A\right)}{1+\sec \left(A\right)}+\frac{1+\sec \left(A\right)}{\tan \left(A\right)}$ is equal to

1. $2\sin A$

2. $2\cos A$

3. $2\cos ecA$

4. $2\sec A$

Q.

$co{t}^{-1}(\sqrt{\cos \alpha )}-{\tan }^{-1}(\sqrt{\cos \alpha )}=x$, then $\sin x$ is

1. ${\tan }^2\left(\frac{\alpha }{2}\right)$

2. $co{t}^2\frac{\alpha }{2}$

3. $\tan \alpha$

4. $cot\frac{\alpha }{2}$

Q.

If the term independent of $x$ in the expansion of ${\left(\frac{3}{2}{x}^2-\frac{1}{3x}\right)}^9$ is $k$, then $18k$ is equal to

1. $5$

2. $9$

3. $7$

4. $11$

Q.

Integration of $\ln xdx$ is

Q.

Write the following functions in the simplest form :
${\tan }^{-1}\left(\frac{\cos x-\sin x}{\cos x+\sin x}\right),0<x<\pi$

Q. The integral $\int \cos \left({\log }_{e}x\right)dx$ is equal to: (where $C$ is a constant of integration)

1. $x\left[\cos \right({\log }_{e}x)+\sin ({\log }_{e}x\left)\right]+C$
2. $x\left[\cos \right({\log }_{e}x)-\sin ({\log }_{e}x\left)\right]+C$
3. $\frac{x}{2}\left[\sin \right({\log }_{e}x)-\sin ({\log }_{e}x\left)\right]+C$
4. $\frac{x}{2}\left[\cos \right({\log }_{e}x)+\sin ({\log }_{e}x\left)\right]+C$

Q.

Prove that $\frac{\tan \theta }{1-cot\theta }+\frac{cot\theta }{1-\tan \theta }=1+sec\theta \cos ec\theta$.

Q.

Find the value of ${\text{sin}}^{-1}\left[\text{cos}\left(\frac{53\mathrm{\pi }}{5}\right)\right]$ is :

Q.

If $x^{2/3}+y^{2/3}=a^{2/3}$, then $\frac{dy}{dx}=$

1. ${\left(\frac{y}{3}\right)}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$3$}\right.}$

2. ${\left(-\frac{y}{x}\right)}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$3$}\right.}$

3. ${\left(\frac{x}{y}\right)}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$3$}\right.}$

4. ${\left(-\frac{x}{y}\right)}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$3$}\right.}$

Q.

If $\int \frac{\cos \theta }{5+7\sin \theta -2{\cos }^2\theta }d\theta =A{\log }_e\left|B\left(\theta \right)\right|+C$ where $C$ is a constant of integration, then $\frac{B\left(\theta \right)}{A}$ can be

1. $\frac{5\left(2\sin \theta +1\right)}{(\sin \theta +3)}$

2. $\frac{5\left(\sin \theta +3\right)}{(2\sin \theta +1)}$

3. $\frac{2\sin \theta +1}{(\sin \theta +3)}$

4. $\frac{2\sin \theta +1}{5(\sin \theta +3)}$

Q. If $f\left(x\right)=\{\begin{array}{c}x\sin \frac{1}{x}\text{else where}\\ 0x=0\end{array},$
then $f\left(x\right)$ is

1. not continuous at $x=0$
2. continuous but non-differentiable at $x=0$
3. differentiable at $x=0$ and $f^{\prime }\left(0\right)=1$
4. differentiable at $x=0$ and $f^{\prime }\left(0\right)=0$

Q. A tangent to the ellipse $x^2+4y^2=4$ meets the ellipse $x^2+2y^2=6$ at $P$ and $Q$. Then the angle between pair of tangents at $P$ and $Q$ of the ellipse $x^2+2y^2=6$ is

1. $\frac{\pi }{6}$
2. $\frac{\pi }{3}$
3. $\frac{\pi }{2}$
4. $\frac{2\pi }{3}$

Q. If $\int x^5{e}^{-4x^3}dx=\frac{1}{48}{e}^{-4x^3}f\left(x\right)+C$, where $C$ is a constant of integration, then $f\left(x\right)$ is equal to :

1. $-4x^3-1$
2. $4x^3+1$
3. $-2x^3-1$
4. $-2x^3+1$

Q.

If $y=a\sin x+bc\mathrm{os}x$, then $y^2+{\left(\frac{dy}{dx}\right)}^2$ is equal to?

1. function of $x$

2. function of $y$

3. function of $x$ and $y$

4. constant

Q.

If $y=\log {\left(\frac{1+x}{1-x}\right)}^{\frac{1}{4}}-\frac{1}{2}{\tan }^{-1}x$ , then $\frac{\mathrm{dy}}{dx}$

1. $\frac{x^2}{\left(1-x^4\right)}$

2. $\frac{2x^2}{\left(1-x^4\right)}$

3. $\frac{x^2}{2\left(1-x^4\right)}$

4. none of these

Q.

$\int \frac{(x^3+3x^2+3x+1)}{{(x+1)}^5}dx=?$

1. $-\frac{1}{(x+1)}+c$

2. $\frac{1}{5}\log (x+1)+c$

3. $\log (x+1)+c$

4. ${\tan }^{-1}x+c$

Q. The integral $\int (1+x-\frac{1}{x})e^{x+\frac{1}{x}}dx$ is equal to:

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

Q.

$\frac{d}{dx}\sqrt{(x\sin x)}=$

1. $\frac{(\sin x+x\cos x)}{2\sqrt{(x\sin x)}}$

2. $\frac{(\sin x+x\cos x)}{\sqrt{(x\sin x)}}$

3. $\frac{(x\sin x+x\cos x)}{2\sqrt{(x\sin x)}}$

4. $\frac{(x\sin x+x\cos x)}{\sqrt{(2x\sin x)}}$

Q.

The dual of the statement $\left[\mathrm{p}\vee \left(~\mathrm{q}\right)\right]\wedge \left(~\mathrm{p}\right)$ is

1. $\text{p∨}\left(~\mathrm{q}\right)\vee ~\mathrm{q}$

2. $\left(\mathrm{p}\wedge ~\mathrm{q}\right)\vee ~\mathrm{p}$

3. $\mathrm{p}\wedge ~\left(\mathrm{q}\vee ~\mathrm{p}\right)$

4. None of the above

Q.

If $f(x+2y,x-2y)=xy$, then $f(x,y)$ is equal to:

1. $\frac{x^2-y^2}{8}$

2. $\frac{x^2-y^2}{4}$

3. $\frac{x^2+y^2}{4}$

4. $\frac{x^2-y^2}{2}$

Q.

The integral $\int \left[\left(2x-1\right)\cos \left(\sqrt{{(2x-1)}^2+5}\right)/\left(\sqrt{4x^2-4x+6}\right)\right]dx$is equal to (where $C$ is a constant of integration)

1. $\left(\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.\right)\sin \left(\sqrt{{(2x+1)}^2+5}\right)+C$

2. $\left(\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.\right)\sin \left(\sqrt{{(2x-1)}^2+5}\right)+C$

3. $\left(\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.\right)\cos \left(\sqrt{{(2x+1)}^2+5}\right)+C$

4. $\left(\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.\right)\cos \left(\sqrt{{(2x-1)}^2+5}\right)+C$

Q.

$\int \frac{1}{\sin (x-a)\sin (x-b)}dx=$

1. $\frac{1}{\sin (b-a)\log \left|{\displaystyle \frac{\sin (x+b)}{\sin (x+a)}}\right|}+C$

2. $\frac{1}{\sin (b+a)\log \left|{\displaystyle \frac{\sin (x-b)}{\sin (x-a)}}\right|}-C$

3. $\frac{1}{\sin (b-a)\log \left|{\displaystyle \frac{\sin (x-a)}{\sin (x-b)}}\right|}+C$

4. None of these

Q.

$\cos 2\theta +2\cos \theta$ is always

1. greater than $-\frac{3}{2}$

2. less than or equal to $\frac{3}{2}$

3. greater than or equal to $-\frac{3}{2}$ and less than or equal to $3$

4. None of these

Q.

Find the general solution of the following .

2sin²x+3cosx=0

Q.

If $\int g\left(x\right)dx=g\left(x\right)$, then $\int g\left(x\right)\left\{f\right(x)+f^{\prime }(x\left)\right\}dx$ is equal to

1. $g\left(x\right)f\left(x\right)-g\left(x\right)f^{\prime }\left(x\right)+C$

2. $g\left(x\right)f^2\left(x\right)+C$

3. $g\left(x\right)f^{\prime }\left(x\right)+C$

4. $g\left(x\right)f\left(x\right)+C$

Q. Suppose $\int \frac{1-7{\cos }^{2}x}{{\sin }^{7}x{\cos }^{2}x}dx=\frac{g\left(x\right)}{{\sin }^{7}x}+C,$ where $C$ is arbitrary constant of integration. Then the value of $g^{\prime }\left(0\right)+g^{\prime \prime }\left(\frac{\pi }{4}\right)$ is

Q.

Evaluate :$\int \frac{x{e}^x}{{(1+x)}^2}dx$

1. $\left[\frac{-e^x}{(x+1)}\right]+c$

2. $\left[\frac{e^x}{(x+1)}\right]+c$

3. $\left[\frac{x{e}^x}{(x+1)}\right]+c$

4. $\left[\frac{-x{e}^x}{(x+1)}\right]+c$