MCQ
$\int \limits_{\frac{3 \sqrt{2}}{4}}^{\frac{3 \sqrt{3}}{4}} \frac{48}{\sqrt{9-4 x^2}} d x$ is equal to
- A$\frac{\pi}{3}$
- B$\frac{\pi}{2}$
- C$\frac{\pi}{6}$
- ✓$2 \pi$
We have $\int \frac{d x}{\sqrt{a^2-x^2}}=\sin ^{-1} \frac{x}{a}+C$
$\text { Hence } \int \limits_{\frac{3 \sqrt{2}}{4}}^{\frac{3 \sqrt{3}}{4}} \frac{48}{\sqrt{9-4 x^2}} dx =\frac{48}{2} \times\left[\sin ^{-1} \frac{2 x}{3}\right]_{\frac{3 \sqrt{2}}{4}}^{\frac{3 \sqrt{3}}{4}}$
$=24 \times\left[\sin ^{-1}\left(\frac{2}{3} \times \frac{3 \sqrt{3}}{4}\right)-\sin ^{-1}\left(\frac{2}{3} \times \frac{3 \sqrt{2}}{4}\right)\right]$
$=24 \times\left[\sin ^{-1} \frac{\sqrt{3}}{2}-\sin ^{-1} \frac{1}{\sqrt{2}}\right]$
$=24 \times\left(\frac{\pi}{3}-\frac{\pi}{4}\right)$
$=24 \times \frac{\pi}{12}=2 \pi$
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Statement $-2:$ The functions $x^2e^x$ and $x^2e^{-x}$ are increasing for all $x > 0$ and the sum of two increasing functions in any interval $(a, b)$ is an increasing function in $(a, b).$
| Column $I$ | Column $II$ |
| $(A)$ $\int_{-1}^1 \frac{\mathrm{dx}}{1+\mathrm{x}^2}$ | $(p)$ $\frac{1}{2} \log \left(\frac{2}{3}\right)$ |
| $(B)$ $\int_0^1 \frac{\mathrm{dx}}{\sqrt{1-\mathrm{x}^2}}$ | $(q)$ $2 \log \left(\frac{2}{3}\right)$ |
| $(C)$ $\int_2^3 \frac{\mathrm{dx}}{1-\mathrm{x}^2}$ | $(r)$ $\frac{\pi}{3}$ |
| $(D)$ $\int_1^2 \frac{d x}{x \sqrt{x^2-1}}$ | $(s)$ $\frac{\pi}{2}$ |