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

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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!...Show more

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...Show more

2 / √x = 2 * x^(-1/2)
Use the power rule of integration...Show more

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