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

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

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

• $100(\mathrm{e} - 1)$

• $\frac{\mathrm{e} - 1}{100}$

If f(x) = ${\int }_{- 1}^{\mathrm{x}}\left|\mathrm{t}\right|d\mathrm{t}$, then for any $\mathrm{x} \ge 0$, f(x) is equal to

• $\frac{1}{2}\left(1 - {\mathrm{x}}^2\right)$

• $\left(1 - {\mathrm{x}}^2\right)$

• $\frac{1}{2}\left(1 + {\mathrm{x}}^2\right)$

• $\left(1 + {\mathrm{x}}^2\right)$

Let I = ${\int }_0^{100\mathrm{\pi }}\sqrt{\left(1 - \cos \left(2\mathrm{x}\right)\right)}d\mathrm{x}$, then

• I = 0

• I = 200$\sqrt{2}$

• I = $\mathrm{\pi }\sqrt{2}$

• I = 100

328.

Let f be a non-constant continuous function for all x > 0. Let f satisfy the relation f(ax) f(a-x)=1 for some a $\in$ R*. Then, I = ${\int }_0^{\mathrm{a}}\frac{\mathrm{dx}}{1 + \mathrm{f}\left(\mathrm{x}\right)}$ is equal to

• a

• $\frac{\mathrm{a}}{4}$

• $\frac{\mathrm{a}}{2}$

• f(a)

$\int \frac{\log \left(\sqrt{\mathrm{x}}\right)}{3\mathrm{x}}\mathrm{dx}$ is equal to

• $\frac{1}{3}{\left(\log \left(\sqrt{\mathrm{x}}\right)\right)}^2 + \mathrm{C}$

• $\frac{2}{3}{\left(\log \left(\sqrt{\mathrm{x}}\right)\right)}^2 + \mathrm{C}$

• $\frac{2}{3}{\left(\log \left(\mathrm{x}\right)\right)}^2 + \mathrm{C}$

• $\frac{1}{3}{\left(\log \left(\mathrm{x}\right)\right)}^2 + \mathrm{C}$

$\int 2^{\mathrm{x}}\left[\mathrm{f}\text{'}\left(\mathrm{x}\right) + \mathrm{f}\left(\mathrm{x}\right)\log \left(2\right)\right]\mathrm{dx}$

• 2^xf'(x) + C

• 2^xlog(2) + C

• 2^xf(x) + C

• 2^x + C