MCQ
A primitive of $f(x) = \frac{x}{{1 + {x^2}}}$ $dx$=
- A${\log _e}({x^2} + 1)$
- B$x{\tan ^{ - 1}}x$
- ✓$\frac{{{{\log }_e}({x^2} + 1)}}{2}$
- D$\frac{1}{2}x{\tan ^{ - 1}}x$
$\Rightarrow 2x\,dx = dt \Rightarrow x\,dx = dt2$
$\therefore \,\,\,I = \frac{1}{2}\int_{}^{} {\frac{{dt}}{t} = \frac{1}{2}\log t + c} $; $I = \frac{1}{2}\log (1 + {x^2}) + c$.
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Statement $-2$ : $f(x) = \frac{1}{{\sqrt {1 - {x^2}} }} + \left[ {\frac{{{x^2} + x + 1}}{4}} \right]$ , where $[.]$ is greatest integer function. Function $f(x)$ is even function
| Column $I$ | Column $II$ |
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$(A)$ Root$(s)$ of the equation $2 \sin ^2 \theta+\sin ^2 2 \theta=2$ |
$(p)$ $\frac{\pi}{6}$ |
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$(B)$ Points of discontinuity of the function $f(x)=\left[\frac{6 x}{\pi}\right] \cos \left[\frac{3 x}{\pi}\right],$ where $[y]$ denotes the largest integer less than or equal to $y$ |
$(q)$ $\frac{\pi}{4}$ |
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$(C)$ Volume of the parallelopiped with its edges represented by the vectors $\hat{i}+\hat{j}, \quad \hat{i}+2 \hat{j} \text { and } \hat{i}+\hat{j}+\pi \hat{k}$ |
$(r)$ $\frac{\pi}{3}$ |
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$(D)$ Angle between vectors $\vec{a}$ and $\vec{b}$ where $\vec{a}, \vec{b}$ and $\vec{c}$ are unit vectors satisfying $\vec{a}+\vec{b}+\sqrt{3} \vec{c}=\overrightarrow{0}$ |
$(s)$ $\frac{\pi}{2}$ |
| $(t)$ $\pi$ |