can you help me integrate this?

Question

integrate: 2/squareroot (1-25x^2)

there are two methods i know:

Using the long method, my answer is 2/5sin^-1 5x + C

Using the short alternative method, my answer is 2sin^-1 5x + C

Which answer is correct? Please justify ur explanation & can you derive a conclusion or something. thx

Answer 1

You can check the answers by taking the derivatives.
The derivative of the first gives the correct result.
The derivative of the second is 10/sqrt(1-25x^2) and hence incorrect.

Answer 2

We know that the derivative of f(x) = arcsin(x) = 1/sqrt(1 - x^2).

1/sqrt(1 - 25x^2) = 1/sqrt(1 - (5x)^2) = (1/5) *5/sqrt(1 - (5x)^2)

Since the derivative of 5x is 5, the integral of 5/sqrt(1 - (5x)^2) is arcsin(5x) + C. So, the integral ypou want is

(2/5) arcsin(5x) + C or, in your notation, (2/5) sin^-1(5x) + C, the first answer is correct, the other wrong.

I don't know what you mean by long and short methods, but any all methods have to lead to theb same solution (except for the constant of integration).