If 

• (sinα, cosα)

• (cosα, sinα)

• (- sinα, cosα)

• (- sinα, cosα)

The value of 

• 28/7

• 28/3

• 7/3

• 7/3

The value of 

• 0

• 1

• 2

• 2

If

• 0

• π

• π/2

• π/2

If  then the value of I₂/I₁ is

• 2

• -2

• 1

• 1

317.

The value of ${\int }_{-\frac{\pi }{2}}^{\frac{\mathrm{\pi }}{2}}\frac{{\sin }^{2}x}{1+2^x}dx is:$

• π/4

• π/8

• π/2

• 4π

The Integral$\int \frac{{\sin }^2 x {\cos }^2 x }{({\sin }^5 x + {\cos }^3 x {\sin }^2 x + {\sin }^3 x {\cos }^2 x + cos{ }^{5}x{)}^2}dx$ is equal to

(where C is a constant of integration)

• $\frac{-1}{1+ co{t}^3 x } + C$

• $\frac{1}{3(1 + {\tan }^3 x) } + C$

• $\frac{-1}{3(1 + {\tan }^3 x )} +C$

• $\frac{1}{1+ co{t}^3 x } + C$

$\int \cos \left(\log \left(\mathrm{x}\right)\right)\mathrm{dx} = \mathrm{F}\left(\mathrm{x}\right) +\hspace{0.17em}\mathrm{C}$, where C is an arbitrary constant. Here, F(x) is equal to 

• $\mathrm{x}\left[\cos \left(\log \left(\mathrm{x}\right)\right) +\hspace{0.17em}\sin \left(\log \left(\mathrm{x}\right)\right)\right]$

• $\mathrm{x}\left[\cos \left(\log \left(\mathrm{x}\right)\right) - \hspace{0.17em}\sin \left(\log \left(\mathrm{x}\right)\right)\right]$

• $\frac{\mathrm{x}}{2}\left[\cos \left(\log \left(\mathrm{x}\right)\right) +\hspace{0.17em}\sin \left(\log \left(\mathrm{x}\right)\right)\right]$

• $\frac{\mathrm{x}}{2}\left[\cos \left(\log \left(\mathrm{x}\right)\right) - \hspace{0.17em}\sin \left(\log \left(\mathrm{x}\right)\right)\right]$

$\int \frac{{\mathrm{x}}^2 - 1}{{\mathrm{x}}^4 + 3{\mathrm{x}}^2 + 1}\mathrm{dx}(\mathrm{x} > 0) \mathrm{is}$

• ${\tan }^{-1}\left(\mathrm{x} + \frac{1}{\mathrm{x}}\right) +\hspace{0.17em}\mathrm{C}$

• ${\tan }^{-1}\left(\mathrm{x} - \frac{1}{\mathrm{x}}\right) +\hspace{0.17em}\mathrm{C}$

• ${\log }_{\mathrm{e}}\left|\frac{\mathrm{x} + {\displaystyle \frac{1}{\mathrm{x}}} - 1}{\mathrm{x} + \frac{1}{\mathrm{x}} + 1}\right| + \mathrm{C}$

• ${\log }_{\mathrm{e}}\left|\frac{\mathrm{x} - {\displaystyle \frac{1}{\mathrm{x}}} - 1}{\mathrm{x} - \frac{1}{\mathrm{x}} + 1}\right| + \mathrm{C}$