You may not know integration , but using dimensional analysis you can check on some results . In the integral `int ( dx)/(( 2ax - x^(2))^(1//2)) = a^(n) sin^(-1) ((x)/(a) -1)`, find the valiue of `n` .

Options

A. `1`
B. `2`
C. `1//2`
D. `3`

Answer

A

Explanation

`int(xdx)/(sqrt(2ax-x^(2)))=a^(n)sin^(-1)[x/a-1]`
L.H.S. is the dimension as
denominators `2ax-x^(2)` must have the dimension of `[x]^(2)`
(` :'` we can add or subtract only it quantities have same dimension)
`:. [sqrt(2ax-x^(2))]=[x]`
Also, dx has the dimension of [x]
`:. (x dx)/(sqrt(2ax-x^(2)))` is having dimesion L
Equating the dimension of L.H.S and R.H.S we have
`[a^(n)]=M^(0)L^(1)T^(0)` ( `:. sin^(-1) (x/a-1)` must be dimensionless)
`:. n=1`