Explain this antiderivative (2/sqrt(x) --> 4sqrt(x) please?

Hi

The power rule states that:

∫ ax^n dx = [a/(n + 1)]x^(n + 1) + C, if n ≠ -1

Notice that:

2/√x = 2/x^(1/2) = 2x^(-1/2)

Then using the power rule, we get:

∫ 2/√x dx

= ∫ 2x^(-1/2) dx

= [2/(-1/2 + 1)]x^(-1/2 + 1) + C

= [2/(1/2)]x^(1/2) + C

= 4x^(1/2) + C

= 4√x + C

You can double check by differentiating 4√x. Since integration is the inverse of differentiation, we should get 2/√x after differentiating 4√x (no need to include the C because the derivative of a constant is 0).

d/dx(4√x)

= d/dx[4x^(1/2)]

= 4(1/2)x^(1/2 - 1)

= 2x^(-1/2)

= 2/√x

So we did it correctly.

I hope this helps!

The antiderivative is the same thing as an integral...

∫2/√x = ∫2/(x^1/2) = 2∫x^(-1/2)

To find the integral (or antiderivative) and a power of x and divide by the new exponent (∫x =x^2/2) So...

2∫x^(-1/2) = 2(x^(1/2)/1/2) which equals 4√x

2 / √x = 2 * x^(-1/2)

Use the power rule of integration

hint: 2/sqrt(x) = 2 / (x^(1/2)) = 2x^(-1/2)