Mathematics : Integrals

$\mathrm{If} \int \frac{\mathrm{dx}}{\mathrm{xlog}\left(\mathrm{x} - 2\right)\log \left(\mathrm{x} - 3\right)} = \mathrm{I} + \mathrm{C}, \mathrm{then} \mathrm{I} =?$

• $\frac{1}{\mathrm{x}}\log \left|\frac{\log \left(\mathrm{x} - 3\right)}{\log \left(\mathrm{x} - 2\right)}\right|$

• $\log \left|\frac{\log \left(\mathrm{x} - 3\right)}{\log \left(\mathrm{x} - 2\right)}\right|$

• $\log \left|\frac{\log \left(\mathrm{x} - 2\right)}{\log \left(\mathrm{x} - 3\right)}\right|$

• $\log \left|\log \left(\mathrm{x} - 3\right)\log \left(\mathrm{x} - 2\right)\right|$

$\mathrm{If} {\int }_0^{\mathrm{b}}\frac{\mathrm{dx}}{1 + {\mathrm{x}}^2} = {\int }_{\mathrm{b}}^{\infty }\frac{\mathrm{dx}}{1 + {\mathrm{x}}^2}, \mathrm{then} \mathrm{b} = ?$

• ${\tan }^{-1}\left(\frac{1}{3}\right)$

• $\frac{\sqrt{3}}{2}$

• $\sqrt{2}$

• 1

The approximate value of ${\int }_1^3\frac{\mathrm{dx}}{2 + 3\mathrm{x}}$ using Simpson's rule and dividing the interval [1, 3] into two equal parts is

• $\frac{1}{3}\log \left(\frac{11}{5}\right)$

• $\frac{107}{110}$

• $\frac{22}{110}$

• $\frac{119}{440}$

$\mathrm{If} \int \frac{\mathrm{dx}}{\sqrt{{\sin }^3\left(\mathrm{x}\right)\cos \left(\mathrm{x}\right)}} = \mathrm{g}\left(\mathrm{x}\right) + \mathrm{c}, \mathrm{then} \mathrm{g}\left(\mathrm{x}\right) = ?$

• $- \frac{2}{\sqrt{\cot \left(\mathrm{x}\right)}}$

• $- \frac{2}{\sqrt{\tan \left(\mathrm{x}\right)}}$

• $\frac{2}{\sqrt{\cot \left(\mathrm{x}\right)}}$

• $\frac{2}{\sqrt{\tan \left(\mathrm{x}\right)}}$

$\mathrm{If} \int \frac{\mathrm{dx}}{\left(1 + \sqrt{\mathrm{x}}\right)\sqrt{\mathrm{x} - {\mathrm{x}}^2}} = \frac{\mathrm{A}\sqrt{\mathrm{x}}}{\sqrt{1 -\mathrm{x}}} + \frac{\mathrm{B}}{\sqrt{1 - \mathrm{x}}} + \mathrm{C}, \mathrm{where} \mathrm{C} \mathrm{is} \mathrm{real} \mathrm{constant}, \mathrm{then} \mathrm{A} + \mathrm{B} = ?$

• 3

• 0

• 1

• 2

For any integer n $\ge$ 2, let I_n = $\int$tanⁿ(x)dx. If  ${\mathrm{I}}_{\mathrm{n}} = \frac{1}{\mathrm{a}}{\tan }^{\mathrm{n} - 1}\left(\mathrm{x}\right) - {\mathrm{bI}}_{\mathrm{n} - 2} \mathrm{for} \mathrm{n}\ge 2,$ then the ordered par (a, b) equals to

• $\left(\mathrm{n} - 1, \frac{\mathrm{n} - 1}{\mathrm{n} - 2}\right)$

• $\left(\mathrm{n} - 1, \frac{\mathrm{n} - 2}{\mathrm{n} - 1}\right)$

• (n, 1)

• (n - 1, 1)

$$\begin{gathered} \mathrm{If} \frac{\left({\mathrm{x}}^2 - 1\right)}{{\left(\mathrm{x} + 1\right)}^2\sqrt{\mathrm{x}\left({\mathrm{x}}^2 + \mathrm{x}\hspace{0.17em}+\hspace{0.17em}1\right)}}\mathrm{dx} \\ = {\mathrm{Atan}}^{-1}\left(\sqrt{\frac{{\mathrm{x}}^2 + \mathrm{x} +\hspace{0.17em}1}{\mathrm{x}}}\right) \mathrm{C}, \mathrm{in} \mathrm{which} \mathrm{C} \mathrm{is} \mathrm{a} \mathrm{constant}, \mathrm{then} \mathrm{A} =? \end{gathered}$$

• $\frac{1}{2}$

• 3

• 2

• 1

$$\begin{gathered} \mathrm{By} \mathrm{the} \mathrm{defination} \mathrm{of} \mathrm{the} \mathrm{definite} \mathrm{integral}, \mathrm{the} \mathrm{value} \mathrm{of} \\ \underset{\mathrm{n}\to \infty }{\lim }\left(\frac{1^4}{1^5 +\hspace{0.17em}{\mathrm{n}}^5} + \frac{2^4}{2^5 + {\mathrm{n}}^5} + \frac{3^4}{3^5 + {\mathrm{n}}^6} + ... + \frac{{\mathrm{n}}^4}{{\mathrm{n}}^5 + {\mathrm{n}}^5}\right) \mathrm{is} \end{gathered}$$

• $\log \left(2\right)$

• $\frac{1}{5}\log \left(2\right)$

• $\frac{1}{4}\log \left(2\right)$

• $\frac{1}{3}\log \left(2\right)$

${\int }_0^{\raisebox{1ex}{$\mathrm{\pi }$}\!\left/ \!\raisebox{-1ex}{$6$}\right.}{\cos }^4\left(3\mathrm{\theta }\right) . {\sin }^2\left(6\mathrm{\theta }\right)\mathrm{d\theta } = ?$

• $\frac{\mathrm{\pi }}{96}$

• $\frac{5}{192}$

• $\frac{5\mathrm{\pi }}{256}$

• $\frac{5\mathrm{\pi }}{192}$

$\int \frac{\mathrm{dx}}{\left(\mathrm{x} - 1\right)\sqrt{{\mathrm{x}}^2 - 1}}$ is equal to

• $- \sqrt{\frac{\mathrm{x} - 1}{\mathrm{x} + 1}} + \mathrm{C}$

• $\sqrt{\frac{\mathrm{x} - 1}{{\mathrm{x}}^2 + 1}} + \mathrm{C}$

• $- \sqrt{\frac{\mathrm{x} + 1}{\mathrm{x} - 1}} + \mathrm{C}$

• $\sqrt{\frac{{\mathrm{x}}^2 + 1}{\mathrm{x} - 1}} + \mathrm{C}$