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Find the value of $\displaystyle\int_{-\pi/2}^{\pi/2}\dfrac{\sin^2 x}{1+2^x}\,dx$
Let $I=\displaystyle\int_{-\pi/2}^{\pi/2}\dfrac{\sin^2x}{1+2^x}dx$. Use the property $\int_{-a}^a f(x)dx$: replace $x\to-x$:
$I=\displaystyle\int_{-\pi/2}^{\pi/2}\dfrac{\sin^2x}{1+2^{-x}}dx=\displaystyle\int_{-\pi/2}^{\pi/2}\dfrac{2^x\sin^2x}{1+2^x}dx$
Adding: $2I=\displaystyle\int_{-\pi/2}^{\pi/2}\sin^2x\,dx=\displaystyle\int_{-\pi/2}^{\pi/2}\dfrac{1-\cos2x}{2}dx=\dfrac{\pi}{2}$
$I=\dfrac{\pi}{4}$
The dimensions of $(\mu_0 \varepsilon_0)^{1/2}$ are:
We know that $c = \dfrac{1}{\sqrt{\mu_0 \varepsilon_0}}$, where $c$ is the speed of light with dimension $[LT^{-1}]$.
Therefore: $\sqrt{\mu_0\varepsilon_0} = \dfrac{1}{c}$
$[\sqrt{\mu_0\varepsilon_0}] = [c^{-1}] = [L^{-1}T]$
So $(\mu_0\varepsilon_0)^{1/2}$ has dimensions $[L^{-1}T]$.
The National Education Policy 1970 was prepared under the leadership of Air Marshal Noor Khan, who served as the Federal Education Minister at the time under the Yahya Khan government. This policy focused on expanding access and improving educational quality.
Correct Answer: C — Noor Khan.
Who authored the work 'On Liberty'?
J.S. Mill's 'On Liberty' is a classic defense of individual freedom from state interference.
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