Solveeit Logo
Login

Question

Question: You may not know integration. But using dimensional analysis you can check on some results. In the i...

You may not know integration. But using dimensional analysis you can check on some results. In the integral ∫dx(2ax−x2)1/2=ansin⁡−1(xa−1)\int_{}^{}{\frac{dx}{(2ax - x^{2})^{1/2}} = a^{n}\sin^{- 1}\left( \frac{x}{a} - 1 \right)}the value of n is

A

1

B

– 1

C

0

D

12\frac{1}{2}

Answer

0

Explanation

Solution

Let x = length ∴[X]=[L]\therefore\lbrack X\rbrack = \lbrack L\rbrack and [dx]⥂=[L]\lbrack dx\rbrack ⥂ = \lbrack L\rbrack

By principle of dimensional homogeneity [xa]=\left\lbrack \frac{x}{a} \right\rbrack =dimensionless ∴[a]=[x]=[L]\therefore\lbrack a\rbrack = \lbrack x\rbrack = \lbrack L\rbrack

By substituting dimension of each quantity in both sides: [L][L2−L2]1/2=[Ln]\frac{\lbrack L\rbrack}{\lbrack L^{2} - L^{2}\rbrack^{1/2}} = \lbrack L^{n}\rbrack ∴n=0\therefore n = 0