Question

Let xand a stand for distance . Is int (dx)/(sqrt (a^(2) - x^(2))) = (1)/(a) sin^(-1) (a/(x)) dimensionally current ?

Answer 1

Let x and a stand for distance . Then int (dx)/(sqrt (a^(2) - x^(2))) = (1)/(a) sin^(-1) (x/(a)) is dimensionally correct .

The RHS of the equation, in the question is wrong.

• ∫dx/√a² - x² = = >

By taking a outside the square root,

• Intergral  =  ∫$\frac{dx}{a\sqrt{1 + \frac{x}{a} ^{2} } }$  = $sin^{-1} \frac{x}{a}$  + c
• Let x/a = u
• Hence dx = adu

• Intergral  = ∫$\frac{du}{\sqrt{1 + u^{2} } }$ = $sin^{-1}$u + c = $sin^{-1} \frac{x}{a}$ + c

If we analyse the dimension of the Integral in LHS, then

• [dx] = [L] ,
• [a] = [L],
• [x/a] = 1
• Therefore, dimension of integral = [1]

Analysing RHS

• Dimension = [$sin^{-1} \frac{x}{a}$] = 1

Therefore they are dimensionally correct.