If  then the value of I₂/I₁ is

• 2

• -2

• 1

• 1

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

33.

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 

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

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

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

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

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

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

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

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

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

Let I = ${\int }_{10}^{19}\frac{\sin \left(\mathrm{x}\right)}{1 + {\mathrm{x}}^8}d\mathrm{x}$, then

• $\left|\mathrm{I}\right| < {10}^{- 9}$

• $\left|\mathrm{I}\right| < {10}^{- 7}$

• $\left|\mathrm{I}\right| < {10}^{- 5}$

• $\left|\mathrm{I}\right| > {10}^{- 7}$

Let I₁ = ${\int }_0^{\mathrm{n}}\left[\mathrm{x}\right]d\mathrm{x}$ and I₂ = ${\int }_0^{\mathrm{n}}\left\{\mathrm{x}\right\}d\mathrm{x}$, where [x] and {x} are integral and fractional parts of x and n $\in$ N - {1}. Then, I₁/I₂ is equal to

• $\frac{1}{\mathrm{n} - 1}$

• $\frac{1}{\mathrm{n}}$

• n

• n - 1

The value of $\underset{\mathrm{n}\to \infty }{\lim }\left[\frac{\mathrm{n}}{{\mathrm{n}}^2 + 1^2} +\frac{{\displaystyle \mathrm{n}}}{{\displaystyle {\mathrm{n}}^2 + 2^2}} + ... +\hspace{0.17em}\frac{1}{2\mathrm{n}}\right]$ is

• $\frac{\mathrm{n\pi }}{4}$

• $\frac{\mathrm{\pi }}{4}$

• $\frac{\mathrm{\pi }}{4\mathrm{n}}$

• $\frac{\mathrm{\pi }}{2\mathrm{n}}$

The value of ${\int }_0^{100}{\mathrm{e}}^{{\mathrm{x}}^2}d\mathrm{x}$

• is less than 1

• is greater than 1

• is less than or equal to 1

• lies in the closed interval [1, e]

${\int }_0^{100}{\mathrm{e}}^{\mathrm{x} - \left[\mathrm{x}\right]}d\mathrm{x}$ is equal to

• $\frac{{\mathrm{e}}^{100} - 1}{100}$

• $\frac{{\mathrm{e}}^{100} - 1}{\mathrm{e} - 1}$

• $100(\mathrm{e} - 1)$

• $\frac{\mathrm{e} - 1}{100}$