Sure, let's calculate the indefinite integral of √4-5x with respect to x. First, let's rewrite √4-5x as (4-5x)^(1/2) to make it easier to work with.
The integral we want to evaluate is ∫ (4-5x)^(1/2) dx.
To solve this, we can do a u-substitution, which is a common method for evaluating integrals. We use a substitution to convert the integral into a simpler form that we can compute directly.
We'll set u equal to the function under the square root, or the term in the parentheses.
That is, u = 4 - 5x.
Additionally, we find the differential of u, du. The differential du is the derivative of the 'u' variable with respect to 'x' times dx. The derivative of 4 - 5x with respect to x is -5. Therefore, du = -5 dx. However, we need to express dx in terms of du, so we substitute dx = du/(-5).
Now, we rewrite the integral in terms of u and du:
∫ (4-5x)^(1/2) dx transforms to -1/5 ∫ u^(1/2) du.
Now this is a simpler integral that we can solve directly.
The antiderivative of u^(1/2) is (2/3)u^(3/2), so -1/5 ∫ u^(1/2) du = -2/15 u^(3/2), where we replaced the 1/5 with 2/15 because 2 times 1/5 is 2/15.
Finally, we substitute u back into the equation, so -2/15 u^(3/2) = -2/15 (4-5x)^(3/2).
And that's the final answer! Therefore, the indefinite integral of the expression √4-5x with respect to x is -2/15 (4-5x)^(3/2), plus an arbitrary constant C, which we typically add in indefinite integral results.
So the final result is: -2/15(4 - 5x)^(3/2) + C.