Question

Find the integral of ln(x) in terms of y.

Options:
A. ∫ ln(x) dx = xy + C
B. ∫ ln(x) dx = y ln(x) + C
C. ∫ ln(x) dx = x²y + C
D. ∫ ln(x) dx = e(ˣʸ) + C

Answer 1

The integral of ln(x) is x ln(x) - x + C, where C is the constant of integration.

Step-by-step explanation:

The integral of ln(x) is given by ∫ ln(x) dx = x ln(x) - x + C, where C is the constant of integration.

To find the integral of ln(x), we use the property that the integral of ln(x) is equal to x ln(x) - x. Let's go through the steps:

1. Start with the integral ∫ ln(x) dx.
2. Apply integration by parts using u = ln(x) and dv = dx.
3. Differentiate u to get du = (1/x) dx and integrate dv to get v = x.
4. Apply the formula for integration by parts: ∫ u dv = uv - ∫ v du.
5. Substitute the values for u, v, and du into the formula: x ln(x) - ∫ x (1/x) dx.
6. Cancel out the x terms in the integral: x ln(x) - ∫ dx.
7. Integrate the remaining term ∫ dx: x ln(x) - x + C, where C is the constant of integration.