Question

Evaluate the integral ∫ ln(x) dx. (Use C for the constant of integration.)

Answer 1

To evaluate ∫ ln(x) dx, use integration by parts by assigning u = ln(x) and dv = dx. Apply the integration by parts formula and simplify the expression to get x ln(x) - x + C as the final answer.

Step-by-step explanation:

To evaluate the integral ∫ ln(x) dx, we can use integration by parts. Let's assume u = ln(x) and dv = dx. Taking the derivative of u, we get du = 1/x dx. Integrating dv, we get v = x.

Using the integration by parts formula: ∫ u dv = uv - ∫ v du, we can substitute in our values: ∫ ln(x) dx = x ln(x) - ∫ x (1/x) dx. Simplifying, we get ∫ ln(x) dx = x ln(x) - ∫ dx = x ln(x) - x + C, where C is the constant of integration.

Answer 2

`\[ \int \ln(x) \,dx = x(\ln(x) - 1) + C \]`

Step-by-step explanation:

The given integral, ∫ ln(x) dx, evaluates to x(ln(x) - 1) + C, where C is the constant of integration. To understand this result, let's consider the integration process. When integrating ln(x) with respect to x, we use integration by parts, which involves selecting parts of the function to differentiate and integrate.

In this case, we let u = ln(x) and dv = dx. After finding du and v, we apply the integration by parts formula, which is ∫ u dv = uv - ∫ v du. Substituting the values, we get x(ln(x) - 1) as the antiderivative. The constant of integration, C, accounts for any constant term that may have been lost during differentiation. Therefore, the final result for the integral is x(ln(x) - 1) + C.