Integration Using Substitution

Q.

What is the integral of $sec\left(x\right)$?

Q.

$\text{∫}\frac{\left(\mathrm{x}+\mathrm{sinx}\right)}{\left(1+\cos \mathrm{x}\right)}\mathrm{dx}=$

Q.

$\frac{d}{dx}\sqrt{(se{c}^{2}x+\cos e{c}^{2}x)}=$

1. $4\cos ec2xcot2x$

2. $-4\cos ec2xcot2x$

3. $4\cos ecxcot2x$

4. None of these

Q.

The length of the longest interval, in which $f\left(x\right)=3\sin \left(x\right)-4{\sin }^{3}x$ is increasing, is

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

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

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

4. $\mathrm{\pi }$

Q.

If $\mathrm{y}={\sec }^{-1}\left(\frac{\left(\sqrt{\mathrm{x}+1}\right)}{\left(\sqrt{\mathrm{x}-1}\right)}\right)+{\sin }^{-1}\left(\frac{\left(\sqrt{\mathrm{x}-1}\right)}{\left(\sqrt{\mathrm{x}+1}\right)}\right)$, then $\frac{dy}{dx}=$

1. $0$

2. $\frac{1}{\sqrt{x+1}}$

3. $1$

4. None of these

Q.

What is $\mathrm{cosec}x$integration.

Q.

If $\alpha$ and $\beta$are the roots of the equations$6x^2-5x+1=0$, then the value of ${\tan }^{-1}\alpha +{\tan }^{-1}\beta$ is

1. $0$

2. $\pi /4$

3. $1$

4. $\pi /2$

Q.

If $cot\theta +cot\left(\frac{\pi }{4}+\theta \right)=2$, then the general value of $\theta$ is

1. $2n\pi \pm \frac{\pi }{6}$

2. $2n\pi \pm \frac{\pi }{3}$

3. $n\pi \pm \frac{\pi }{3}$

4. $n\pi \pm \frac{\pi }{6}$

Q.

What is the derivative of ${\sin }^{2}x$ with respect of ${\cos }^{2}x$?

1. $\tan x$

2. ${\tan }^{2}x$

3. $-\tan x$

4. None of these

Q.

If $f:R⟶S$ defined by $f\left(x\right)=\sin x-\sqrt{3}\cos x+1$ is onto, then the interval of $S$ is

1. $\left[0,3\right]$

2. $\left[-1,1\right]$

3. $\left[0,1\right]$

4. $\left[-1,3\right]$

Q.

What is the exact value of $\sec \left(\frac{\mathrm{\pi }}{4}\right)$?

Q.

If the vectors $2i-3j+4k$ and $i+2j-k$ and $mi-j+2k$ are coplanar, then the value of $m$ is?

1. $\frac{5}{8}$

2. $\frac{8}{5}$

3. $-\frac{7}{4}$

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

Q.

If $y={\cot }^{-1}\left(\frac{1+x}{1-x}\right)$, then $\frac{dy}{dx}$ equals to?

1. $\frac{1}{1+x^2}$

2. $\frac{-1}{1+x^2}$

3. $\frac{2}{1+x^2}$

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

Q.

$\frac{d\left[{\sin }^n\left(x\right)\cos \left(nx\right)\right]}{dx}=$

1. $n{\sin }^{n-1}\left(x\right)\cos \left(\left(n+1\right)x\right)$

2. $n{\sin }^{n-1}\left(x\right)\cos \left(nx\right)$

3. $n{\sin }^{n-1}\left(x\right)\cos \left(\left(n-1\right)x\right)$

4. $n{\sin }^{n-1}\left(x\right)\sin \left(\left(n+1\right)x\right)$

Q.

$\tan \left[\frac{\mathrm{\pi }}{4}+\left(\frac{1}{2}\right){\cos }^{-1}\left(\frac{a}{b}\right)\right]+\tan \left[\frac{\mathrm{\pi }}{4}-\left(\frac{1}{2}\right){\cos }^{-1}\left(\frac{a}{b}\right)\right]$ is equal to

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

2. $\frac{2b}{a}$

3. $\frac{a}{b}$

4. $\frac{b}{a}$

Q.

If $f\left(x\right)={\int }_{}\frac{5x^8+7x^6}{{\left[x^2+1+2x^7\right]}^2}dx,x\ge 0,f\left(0\right)=0$ and $f\left(1\right)=\frac{1}{k}$, then the value of $k$ is

Q.

The expression $\cos \frac{10\mathrm{\pi }}{13}+\cos \frac{8\mathrm{\pi }}{13}+\cos \frac{3\mathrm{\pi }}{13}+\cos \frac{5\mathrm{\pi }}{13}$ is

1. $-1$

2. $0$

3. $1$

4. None of these

Q.

If for $\mathrm{x}\in \left(0,\frac{\mathrm{\pi }}{2}\right),{\log }_{10}\sin \mathrm{x}+{\log }_{10}\cos \mathrm{x}=-1$ and ${\log }_{10}(\sin \mathrm{x}+\cos \mathrm{x})=\frac{1}{2}({\log }_{10}\mathrm{n}-1),\mathrm{n}>0$, then the value of $n$ is equal to:

1. 16

2. 20

3. 12

4. 9

Q.

If$(a+\sqrt{2}b\cos x)(a-\sqrt{2}b\cos y)=a^2-b^2$, where $a>b>0$, then $dy/dx$ at $(\pi /4,\pi /4)$ is

1. $\frac{(a+b)}{(a-b)}$

2. $\frac{(a-2b)}{(a+2b)}$

3. $\frac{(a-b)}{(a+b)}$

4. $\frac{(2a+b)}{(2a-b)}$

Q.

If $y={\cos }^{–1}(\cos x)$, then $\frac{dy}{dx}$ is equal to ?

1. $1$ in the whole plane

2. $–1$ in the whole plane

3. $1$ in the $2^{nd}$and $3^{rd}$ quadrants of the plane

4. $–1$ in the $3^{rd}$ and $4^{th}$ quadrants of the plane

Q. If $y=x^{(\ln x{)}^{\ln \left(\ln x\right)}},$ then $\frac{dy}{dx}=$

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

Q.

Prove that $\frac{\cos \theta }{(1+sin\theta )}+\frac{(1+\sin \theta )}{cos\theta }=2sec\theta$

Q.

$\underset{x\to 0}{\lim }\frac{\sin \left(\mathrm{\pi }{\cos }^2\mathrm{x}\right)}{x^2}$

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

2. $1$

3. $-\mathrm{\pi }$

4. $\mathrm{\pi }$

Q.

The value of ${\cos }^3\left(\frac{\pi }{8}\right)·\cos \left(\frac{3\pi }{8}\right)+{\sin }^3\left(\frac{\pi }{8}\right)\sin \left(\frac{3\pi }{8}\right)$ is

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

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

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

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

Q. The area bounded by the curves $y=\ln x$ and $y=(\ln x{)}^2$ is

1. $(3-e)\text{sq. units}$
2. $(3+e)\text{sq. units}$
3. $3\text{sq. units}$
4. $(3+2e)\text{sq. units}$

Q.

The value of $\underset{x\to 0}{\lim }\frac{\left[1-\cos (1-\cos x)\right]}{x^4}$ is

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

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

3. $\frac{1}{6}$

4. $\frac{1}{8}$

Q.

If $\alpha =x(a\times b)+y(b\times c)+z(c\times a)$ and $[abc]=\frac{1}{8}$ then $x+y+z$ is equal to?

1. $8\alpha ·(a+b+c)$

2. $\alpha ·(a+b+c)$

3. $8(a+b+c)$

4. None of these

Q.

Find the general solution $\tan x=-1/\surd 3$.

Q. Find the maximum and minimum values of each of the following trigonometrical expressions:
(i) $12sin\theta -5cos\theta$
(ii) $12cos\theta +5sin\theta +4$
(iii) $5cos\theta +3sin(\frac{\pi }{6}-\theta )+4$
(iv) $sin\theta -cos\theta +1$

Q.

If $3\sin \theta +5\cos \theta =5$, the value of $5\sin \theta -3\cos \theta$ is