evaluate
¡Ò (Ln rad(x)) /x dx=
Evaluate the indefinite integral
ln √x = 1/2 ln x
Note that if u = ln x,
du = 1/x dx
so your integral is just 1/2 u du
To evaluate the integral ∫ (Ln rad(x)) / x dx, we can use integration by parts.
Let's start by using the formula for integration by parts: ∫ u dv = uv - ∫ v du.
In this case, we can choose u = Ln rad(x) and dv = dx. Then, we find du and v.
Taking the derivative of u with respect to x, we get:
du/dx = (1/x) · (1/(2·rad(x))) · (1/x)
= 1/(2x^2 · rad(x))
Integrating dv, we get:
v = x
Now, let's substitute u, du, v, and dv into the formula for integration by parts:
∫ (Ln rad(x)) / x dx = u · v - ∫ v · du
= Ln rad(x) · x - ∫ x · 1/(2x^2 · rad(x)) dx
= x Ln rad(x) - (1/2) ∫ 1/(x · rad(x)) dx
The integral ∫ 1/(x · rad(x)) dx can be simplified by using a substitution. Let's substitute u = rad(x). Then, du = (1/(2 · rad(x))) dx and dx = 2 · rad(x) du.
Substituting these into the integral, we have:
(1/2) ∫ 1/(x · rad(x)) dx = (1/2) ∫ 1/(u^2) · 2 · rad(x) du
= ∫ 1/u^2 du
= -1/u + C,
where C is the constant of integration.
Therefore, the final result of the integral is:
∫ (Ln rad(x)) / x dx = x Ln rad(x) - (1/2)(-1/u) + C
= x Ln rad(x) + (1/2)(1/rad(x)) + C.
Note that rad(x) represents the square root of x.