Integration of Trigonometric Functions

Q.

The angles of a triangle are in the ratio $3:5:10$. Then, the ratio of the smallest side to the greatest side is

1. $1:\sin {10}^{°}$

2. $1:2\sin {10}^{°}$

3. $1:\cos {10}^{°}$

4. $1:2\cos {10}^{°}$

Q.

The value of ${\int }_0^1{\tan }^{-1}\left(\frac{2x-1}{1+x-x^2}\right)dx$ is

1. $1$

2. $0$

3. $-1$

4. None of these

Q. $\int \frac{1}{4{\sin }^{2}x+9{\cos }^{2}x}dx$ will be equal to -

1. $\frac{1}{3}{\tan }^{-1}\left(\frac{2\tan \left(x\right)}{3}\right)+C$
2. $\frac{1}{6}{\tan }^{-1}\left(\frac{2\tan \left(x\right)}{3}\right)+C$
3. $\frac{1}{6}{\tan }^{-1}\left(\frac{2\tan \left(x\right)}{6}\right)+C$
4. $\frac{1}{6}{\tan }^{-1}\left(\frac{2\tan \left(x\right)}{5}\right)+C$

Q.

The maximum value of $f\left(x\right)=2\sin x+\cos 2x,0\le x\le \frac{\mathrm{\pi }}{2}$ occurs at $x$is

1. $0$

2. $\frac{\mathrm{\pi }}{6}$

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

4. None of these

Q. If $\underset{0}{\overset{\frac{\pi }{2}}{\int }}\frac{\cot x}{\cot x+\text{cosec}x}dx=m(\pi +n),$ then $m.n$ is equal to :

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

Q.

If $0<\beta <\alpha <\frac{\pi }{2},\cos (\alpha +\beta )=\frac{3}{5}and\cos (\alpha -\beta )=\frac{4}{5}$, then $\sin 2\alpha$ is equal to.

1. $0$

2. $1$

3. $2$

4. $Noneofthese$

Q.

The derivative of $\cos 3x$ with respect to $\sin 3x$ is

Q. If $\int \frac{x^4+1}{x^6+1}dx={\tan }^{-1}f\left(x\right)-\frac{2}{3}{\tan }^{-1}g\left(x\right)+c,$ where $c$ is an arbitrary constant, then

1. both $f\left(x\right)$ and $g\left(x\right)$ are odd functions
2. both $f\left(x\right)$ and $g\left(x\right)$ are even functions
3. $f\left(x\right)=g\left(x\right)$ has no real roots
4. $\int \frac{f\left(x\right)}{g\left(x\right)}dx=\frac{-1}{x}+\frac{1}{3x^3}+d,$ where $d$ is an arbitrary constant.

Q.

Evaluate: $\underset{x\to 0}{\lim }\frac{\sin \left|x\right|}{x}$

1. $1$

2. $0$

3. positive infinity

4. does not exist

Q. If $\int \sqrt{1+\text{cosec}x}dx=f\left(x\right)+C$, where $C$ is a constant of integration, then $f\left(x\right)$ is equal to

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

Q. The value of $\frac{\sin {70}^{\circ }-\cos {40}^{\circ }}{\cos {70}^{\circ }-\sin {40}^{\circ }}$ is equal to

$\frac{\sin {70}^{\circ }-\cos {40}^{\circ }}{\cos {70}^{\circ }-\sin {40}^{\circ }}$ का मान बराबर है

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

Q.

$\int \frac{1}{2+3sinx}dx$

1. $\frac{2}{\sqrt{5}}.log\left|\frac{tan\frac{x}{2}+\frac{3}{2}-\sqrt{\frac{5}{4}}}{tan\frac{x}{2}+\frac{3}{2}+\sqrt{\frac{5}{4}}}\right|+C$

2. $\frac{1}{2\sqrt{5}}.log\left|\frac{tan\frac{x}{2}+\frac{3}{2}-\sqrt{\frac{5}{4}}}{tan\frac{x}{2}+\frac{3}{2}+\sqrt{\frac{5}{4}}}\right|+C$

3. $\frac{3}{\sqrt{5}}.log\left|\frac{tan\frac{x}{2}+\frac{3}{2}-\sqrt{\frac{5}{4}}}{tan\frac{x}{2}+\frac{3}{2}+\sqrt{\frac{5}{4}}}\right|+C$

4. $\frac{1}{\sqrt{5}}.log\left|\frac{tan\frac{x}{2}+\frac{3}{2}-\sqrt{\frac{5}{4}}}{tan\frac{x}{2}+\frac{3}{2}+\sqrt{\frac{5}{4}}}\right|+C$

Q.

If ${\int }_0^{\alpha }\frac{dx}{1-\cos \alpha \cos x}=\frac{A}{\sin \alpha }+B(\alpha \ne 0)$ the values of A and B are

1. $A=\frac{\pi }{2},B=0$ or $A=\frac{\pi }{4},B=\frac{\pi }{4sin\alpha }$

2. $A=\frac{\pi }{4},B=\frac{\pi }{sin\alpha }$

3. $A=\frac{\pi }{6},B=\frac{\pi }{sin\alpha }$

4. $A=\pi ,B=\frac{\pi }{sin\alpha }$

Q. If $\int {\sin }^{-1}\left(\sqrt{\frac{x}{1+x}}\right)dx=A\left(x\right){\tan }^{-1}\left(\sqrt{x}\right)+B\left(x\right)+C,$ where $C$ is a constant of integration, then the ordered pair $\left(A\right(x),B(x\left)\right)$ can be:

1. $(x-1,-\sqrt{x})$
2. $(x+1,\sqrt{x})$
3. $(x+1,-\sqrt{x})$
4. $(x-1,\sqrt{x})$

Q. The value of $\int \frac{x-\sin x}{1-\cos x}dx$ is equal to
(where $C$ is integration constant)

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

Q. $\int \frac{\mathrm{dx}}{{\sin }^{2}x-12{\cos }^{2}x+\mathrm{cosx}\mathrm{sinx}}$ is equal to

1. $\frac{1}{7}\ln \left|\frac{\tan x+3}{\tan x-4}\right|+C$
2. $\frac{1}{7}\ln \left|\frac{\tan x-3}{\tan x+4}\right|+C$
3. $\frac{1}{7}\ln \left|\frac{\tan x+3}{\tan x+4}\right|+C$
4. $\frac{1}{7}\ln \left|\frac{\tan x-3}{\tan x-4}\right|+C$

Q. $\int \frac{\cos x+\sqrt{3}}{1+4\sin (x+\frac{\pi }{3})+4{\sin }^2(x+\frac{\pi }{3})}dx$ is
where $c$ is constant of integration

1. $\frac{\cos x}{1+2\sin (x+\frac{\pi }{3})}+c$
2. $\frac{\sec x}{1+2\sin (x+\frac{\pi }{3})}+c$
3. $\frac{\sin x}{1+2\sin (x+\frac{\pi }{3})}+c$
4. $\frac{1}{2}{\tan }^{-1}(1+2\sin (x+\frac{\pi }{3}))+c$

Q. $\int \frac{co{s}^{4}xdx}{si{n}^{3}x(si{n}^{5}x+co{s}^{5}x{)}^{\frac{3}{5}}}=-\frac{1}{2}(1+co{t}^{A}x{)}^B+C$ then AB=

1. 1
2. 2
3. $\frac{1}{2}$
4. None of these

Q. $\text{If}\int \frac{1}{(\sec x+\text{cosec}x+\tan x+\cot x{)}^2}dx=\frac{x}{\alpha }+\frac{\cos (x+\frac{\pi }{4})}{\beta }+\frac{\cos 2x}{\gamma }+c,\text{then}|\alpha +\sqrt{2}\beta +\gamma |\text{is}$ , (where $c$ is an arbitrary constant)