Integration of Trigonometric Functions

Q. $\int \frac{\text{dx}}{{\text{sin}}^4\text{x}+{\text{cos}}^4\text{x}}$ is equal to

1. $\frac{1}{\sqrt{2}}ta{n}^{-1}\left(\frac{1}{\sqrt{2}}tan2x\right)+C$
2. $\sqrt{2}ta{n}^{-1}\left(\frac{1}{\sqrt{2}}tan2x\right)+C$
3. $\frac{1}{\sqrt{2}}ta{n}^{-1}\left(\frac{1}{\sqrt{2}}cot2x\right)+C$
4. None of these

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. 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. $\int \frac{x+sinx}{1+cosx}dx=$

1. $-log|1+cosx|+C$
2. $xcot\left(\frac{x}{2}\right)+C$
3. $log|1+cosx|+C$
4. $xtan\frac{x}{2}+C$

Q. $\int \frac{(2+secx)secx}{(1+2secx{)}^2}dx$

1. $2cosecx-cotx+C$
   
2. $\frac{1}{2cosecx+cotx}+C$
3. $\frac{1}{2cosecx-cotx}+C$
4. $2cosecx+cotx+C$

Q. If $\int \frac{\tan x}{1+\tan x+{\tan }^{2}x}dx=x-\frac{2}{\sqrt{A}}{\tan }^{-1}\left(\frac{2\tan x+1}{\sqrt{A}}\right)+C,$ where $C$ is arbitrary constant of integration, then the value of $A$ is

Q.

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

1. 

2. 

3. 

4. 

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.

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. $A=\frac{\pi }{2},B=0$

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

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

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

Q. $\int \frac{\mathrm{dx}}{4+5{\sin }^{2}x}\mathrm{dx}$ is equal to.

1. 
   $\frac{1}{6}{\tan }^{-1}\left(\frac{3\mathrm{cotx}}{2}\right)+C$
2. 
   $\frac{1}{6}{\tan }^{-1}\left(\frac{3\mathrm{tanx}}{2}\right)+C$
3. 
   $\frac{1}{6}{\cot }^{-1}\left(\frac{3\mathrm{cotx}}{2}\right)+C$
4. 
   $\frac{1}{6}{\cot }^{-1}\left(\frac{3\mathrm{tanx}}{2}\right)+C$