evaluate the indefinite integral
integral sign [8/x ln(1x)] dx
= +c
To evaluate the indefinite integral of the given function, we can apply the technique of integration by parts. The formula for integration by parts is:
∫ u * v dx = u * ∫ v dx - ∫ (u' * ∫ v dx) dx
In this formula, u and v represent functions, and u' is the derivative of u with respect to x.
In the given integral, let's choose:
u = ln(x)
v = 8
Using the formula for integration by parts, we have:
∫ (8/x * ln(x)) dx = u * ∫ v dx - ∫ (u' * ∫ v dx) dx
Substituting the chosen values:
= ln(x) * ∫ 8 dx - ∫ (1/x * ∫ 8 dx) dx
Now, let's evaluate the two integrals separately:
∫ 8 dx = 8x + C1, where C1 is the constant of integration
∫ (1/x * ∫ 8 dx) dx
Using the same technique, we can choose:
u = 1/x
v = 8x
∫ (1/x * ∫ 8 dx) dx = (1/x) * (8x^2/2) - ∫ [(1/x)' * (8x^2/2)] dx
= 4x - ∫ (8x/x^2) dx
= 4x - ∫ (8/x) dx
= 4x - 8 ln(x) + C2, where C2 is the constant of integration
Putting it all together, we have:
∫ (8/x * ln(x)) dx = ln(x) * (8x + C1) - (4x - 8 ln(x) + C2)
This simplifies to:
∫ (8/x * ln(x)) dx = 8x ln(x) + C1 ln(x) - 4x + 8 ln(x) + C2
Finally, we can combine the constant terms:
∫ (8/x * ln(x)) dx = (8x + 8 ln(x) - 4x) + (C1 ln(x) + C2)
So, the indefinite integral of the function (8/x * ln(x)) dx is:
∫ (8/x * ln(x)) dx = 4x + 8 ln(x) + C, where C is the combined constant of integration (C = C1 ln(x) + C2).