Question
Prove that: $\int \frac{d x}{\sqrt{x^2+a^2}}=\log \left|x+\sqrt{x^2+a^2}\right|+c$
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perpendicular to vectors $\bar{b}=\hat{i}+2 \hat{j}+\hat{k}$ and $\bar{c}=3 \hat{i}+2 \hat{j}-\hat{k}$.
to lines $\frac{x-1}{1}=\frac{y-2}{2}=\frac{z-3}{3}$ and $\frac{x}{-3}=\frac{y}{2}=\frac{z}{5}$.
$\int_0^4 x^2 d x$
$\tan ^{-1}\left(\frac{3 x^2-4 y^2}{3 x^2+4 y^2}\right)=a^2$
$\left(x^2+3\right)^{\frac{3}{2}} \cdot \sin ^3 2 x \cdot 2^{x^2}$
and water is poured into it. If at any instant the water level rises at the rate of $\left(\frac{\pi}{A}\right) cm / sec$,
where A is the area of the water surface at that instant, show that the vessel will be full in 75 seconds.
