Question:
Let $\mathrm{x}$ and a stand for distance. Is
$\int \frac{d x}{\sqrt{a^{2}-x^{2}}}=\frac{1}{a} \sin ^{-1} \frac{a}{x}$ dimensionally correct?
Solution:
Dimension of the Integral $=\int \frac{d x}{\sqrt{a^{2}-x^{2}}}=\int \frac{L}{\sqrt{L^{2}-L^{2}}}=L^{0}$. But for $\frac{1}{a} \sin ^{-1} \frac{a}{x}$, the dimension is [L-1]. So
this expression is dimensionally incorrect.
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