Let x and a stand for distance. Is ∫ dx√a 2 − x 2= 1a sin − 1 ax dimensionally correct ?

To check if the equation is dimensionally correct, we need to ensure that both sides of the equation have the same dimensions.

The left side of the equation is ∫ dx√a 2 − x 2. The integral of dx is x, and the square root of a^2 - x^2 will also have the dimensions of distance, since both a and x represent distance. Therefore, the overall dimensions of the left side of the equation are distance.

The right side of the equation is 1/a * sin^−1(ax). The inverse sine function, sin^−1(ax), is dimensionless, since it represents an angle. The term 1/a has the dimensions of 1/distance. Therefore, the overall dimensions of the right side of the equation are 1/distance.

Since the dimensions of the left side (distance) do not match the dimensions of the right side (1/distance), the equation is not dimensionally correct.

To check if the equation is dimensionally correct, we need to ensure that both sides of the equation have the same dimensions.

The left side of the equation is an integral of dx/√(a^2 - x^2). The dimension of this expression is the dimension of distance (since x is a distance) divided by the square root of the difference of two squares of distances, which is also a distance. So, the dimension of the left side is [L] (where [L] stands for the dimension of length or distance).

The right side of the equation is (1/a)sin^(-1)(x/a). The dimension of this expression is the dimension of the inverse of distance times the dimensionless quantity sin^(-1)(x/a). So, the dimension of the right side is also [L].

Since both sides of the equation have the same dimensions, the equation is dimensionally correct.

To check if the given equation is dimensionally correct, we need to ensure that both sides of the equation have the same dimensions.

Let's break down the dimensions on both sides:

1. On the left side, we have ∫ dx√(a^2 - x^2). The integral of dx is x, and x is a distance, so its dimension is L (length). The square root of (a^2 - x^2) also has dimensions of L, because it's a difference of squares of distances. So, the overall dimension on the left side is L^2.

2. On the right side, we have (1/a)sin^−1(ax). The sine inverse of any quantity is dimensionless, because it's a ratio. Therefore, the dimension on the right side is 1/L.

Since the dimensions on both sides are not the same (L^2 ≠ 1/L), the given equation is not dimensionally correct.