MCQ
$\int_0^2 {\frac{{{x^3}\,dx}}{{{{({x^2} + 1)}^{\frac{3}{2}}}}}} = $
- A${(\sqrt 2 - 1)^2}$
- B$\frac{{{{(\sqrt 2 - 1)}^2}}}{{\sqrt 2 }}$
- C$\frac{{\sqrt 2 - 1}}{{\sqrt 2 }}$
- ✓None of these
$\int_0^2 {\frac{{{x^3}}}{{{{({x^2} + 1)}^{3/2}}}}dx = \frac{1}{2}} \int_1^5 {\frac{{(t - 1)}}{{{t^{3/2}}}}dt = \frac{1}{2}\int_1^5 {[{t^{ - 1/2}} - {t^{ - 3/2}}]\,dt} } $
$ = \frac{1}{2}\left[ {2\sqrt t + 2\frac{1}{{\sqrt t }}} \right]_1^5 $
$= \frac{1}{2}\left[ {2\sqrt 5 + \frac{2}{{\sqrt 5 }} - 2 - 2} \right]$
$ = \left[ {\sqrt 5 + \frac{1}{{\sqrt 5 }} - 2} \right] $
$= \frac{{6 - 2\sqrt 5 }}{{\sqrt 5 }}$.
Generate a complete, print-ready paper with questions like this in minutes — across 16+ boards, with answer keys.
Statement $-1$ : If the graphs of two linear equations in two variables are neither parallel nor the same, then there is a unique solution to the system. Statement $-2$ : If the system of equations $ax + by = 0, cx + dy = 0$ has a non-zero solution, then it has infinitely many solutions.
Statement $-3$ : The system $x + y + z = 1, x = y, y = 1 + z$ is inconsistent. Statement $-4$ : If two of the equations in a system of three linear equations are inconsistent, then the whole system is inconsistent.
$f(x)=\min \{x-[x], 1+[x]-x\}$
where $[\mathrm{x}]$ is the greatest integer less than or equal to $\mathrm{x}$. Let $\mathrm{P}$ denote the set containing all $x \in[0,3]$ where $f$ is discontinuous, and $Q$ denote the set containing all $x \in(0,3)$ where $f$ is not differentiable. Then the sum of number of elements in $\mathrm{P}$ and $\mathrm{Q}$ is equal to $......$