Integration of Trigonometric Functions

Q.

Find the integration of $\sin x\cos xdx$.

Q.

Prove that:

$\frac{\tan \theta }{(1-cot\theta )}+\frac{cot\theta }{(1-\tan \theta )}=1+\tan \theta +cot\theta$

Q. Solve $\sin 2x-\sin 4x+\sin 6x=0$

Q.

Find the value of $9se{c}^{2}a–9{\tan }^{2}a$

Q.

If $\sin \theta =\sin \left(15°\right)+\sin \left(45°\right)$, where $0°<\theta <90°$, then $\theta$ is equal to

1. $45°$

2. $54°$

3. $60°$

4. $72°$

5. $75°$

Q. $\text{If}f\left(x\right)=|\cos x-\sin x|$, then $f^{\prime }\left(\frac{\pi }{6}\right)$ is equal to

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

Q.

The maximum and minimum values of the function $\left|\sin \left(4x\right)+3\right|$ is

1. $1$, $2$

2. $4$, $2$

3. $2$, $4$

4. $-1$, $1$

Q.

If $z=\frac{1+i\sqrt{3}}{\sqrt{3}+i}$, then ${\overline{z}}^{100}$ lies in the

1. first quadrant

2. second quadrant

3. third quadrant

4. fourth quadrant

Q.

$\frac{d}{dx}\left[{\sin }^2\left({\cot }^{-1}\left(\sqrt{\frac{1+x}{1-x}}\right)\right)\right]=$

1. $0$

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

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

4. $-1$

Q.

If ${\tan }^{-1}x+{\tan }^{-1}y=\frac{\pi }{4}$, then

1. $x+y+xy=1$

2. $x+y-xy=1$

3. $x+y+xy+1=0$

4. $x+y-xy+1=0$

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. If $f\left(x\right)=\int e^x(\frac{2}{1-\tan x}+{\tan }^2(x+\frac{\pi }{4}\left)\right)dx$, where $f\left(\frac{3\pi }{4}\right)=0.$ Then the value of $\ln \left(f\right(\pi \left)\right)$ is

Q. If cosA+sinB=m and sinA+cosB=n, prove that $2sin(A+B)=m^2+n^2-2$.

Q.

$\int \left[f\right(x\left)g^{\prime \prime }\right(x)-f"(x\left)g\right(x\left)\right]dx$ is equal to

1. $\frac{f\left(x\right)}{g^{\prime }\left(x\right)}$

2. $f^{\prime }\left(x\right)g\left(x\right)-f\left(x\right)g^{\prime }\left(x\right)$

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

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

Q. If $\int \frac{d\theta }{\left({\cos }^2\theta \right(\tan 2\theta +\sec 2\theta )}=\lambda \tan \theta +2{\log }_e|f\left(\theta \right)|+C$ where $C$ is constant of integration, then the ordered pair $(\lambda ,f(\theta \left)\right)$ is equal to:

1. $(-1,1-\tan \theta )$
2. $(-1,1+\tan \theta )$
3. $(1,1+\tan \theta )$
4. $(-1,1-\tan \theta )$

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.

Prove the following;

$2si{n}^{-1}\frac{3}{5}=ta{n}^{-1}\frac{24}{7}$

Q.

The product of all values of ${\left(\cos \alpha +\iota \sin \alpha \right)}^{\frac{3}{5}}$

1. $1$

2. $\cos \left(\alpha \right)+i\sin \left(\alpha \right)$

3. $\cos \left(3\alpha \right)+i\sin \left(3\alpha \right)$

4. $\cos \left(5\alpha \right)+i\sin \left(\alpha \right)$

Q. The equation of the tangent to the curve
$y={\sin }^{-1}\frac{2x}{1+x^2}$ at $x=\sqrt{3}$

1. $y=-\frac{x}{2}$
2. $y-\frac{\pi }{3}=-\frac{x}{2}$
3. $y-\frac{\pi }{3}=\frac{1}{2}(x-\sqrt{3})$
4. $y-\frac{\pi }{3}=\frac{-1}{2}(x-\sqrt{3})$

Q.

Prove that:
$\tan \left(\frac{\pi }{4}+\theta \right)-\tan \left(\frac{\pi }{4}-\theta \right)=2\tan 2\theta$.

Q.

$\int \frac{dx}{(\sin x-\cos x+\sqrt{2})}=$

1. $\left(\frac{-1}{\sqrt{2}}\right)\tan \left[\left(\frac{x}{2}\right)+\left(\frac{\pi }{8}\right)\right]+c$

2. $\left(\frac{1}{\sqrt{2}}\right)\tan \left[\left(\frac{x}{2}\right)+\left(\frac{\pi }{8}\right)\right]+c$

3. $\left(\frac{1}{\sqrt{2}}\right)\text{cot}\left[\left(\frac{x}{2}\right)+\left(\frac{\pi }{8}\right)\right]+c$

4. $\left(\frac{-1}{\sqrt{2}}\right)\text{cot}\left[\left(\frac{x}{2}\right)+\left(\frac{\pi }{8}\right)\right]+c$

Q. If ${\theta }_1$ and ${\theta }_2$ be respectively the smallest and the largest values of $\theta$ in $(0,2\pi )-\left\{\pi \right\}$ which satisfy the equation, $2{\cot }^2\theta -\frac{5}{\sin \theta }+4=0$, then $\underset{{\theta }_1}{\overset{{\theta }_2}{\int }}{\cos }^{2}3\theta d\theta$ is equal to :

1. $\frac{2\pi }{3}$
2. $\frac{\pi }{3}$
3. $\frac{\pi }{3}+\frac{1}{6}$
4. $\frac{\pi }{9}$

Q. If $\int \frac{cos8x+1}{tan2x-cot2x}dx=acos8x+C$, then

1. $a=\frac{-1}{16}$
2. $a=\frac{1}{8}$
3. $a=\frac{1}{16}$
4. $a=\frac{-1}{8}$

Q. Find the principal value of ${\cos }^{-1}(-\frac{1}{2}).$

Q.

If $3{\sin }^{-1}\left(\frac{2x}{1+x^2}\right)-4{\cos }^{-1}\left(\frac{1-x^2}{1+x^2}\right)+2{\tan }^{-1}\left(\frac{2x}{1-x^2}\right)=\frac{\pi }{3}$, then$x=$

1. $\sqrt{3}$

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

3. $1$

4. None of these

Q. The value of the integral $\int \frac{1}{x^4-1}dx$ is
(where $C$ is an arbitrary constant)

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

Q. If $\Phi \left(x\right)=\int \frac{dx}{si{n}^{\frac{1}{2}}xco{s}^{\frac{7}{2}}x}$, then $\Phi \left(\frac{\pi }{4}\right)-\Phi \left(0\right)=$

1. 0
2. $\frac{12}{5}$
3. $\frac{6}{5}$
4. $\frac{9}{5}$

Q. Find the integral: $\int (2x-3\cos x+e^x)dx$

Q.

If $y=\frac{\tan x+\cot x}{\tan x-\cot x}$, then $\frac{dy}{dx}=$

1. $2\tan 2x\sec 2x$

2. $\tan 2x\sec 2x$

3. $-\tan 2x\sec 2x$

4. $-2\tan 2x\sec 2x$

Q. The length of tangent to the curve $x=a(\cos t+\log \left(\tan \frac{t}{2}\right)),y=a\sin t,$ at any point is :

1. $\left|a\right|$
2. $\left|ax\right|$
3. $\left|ay\right|$
4. $xy$