Evaluate ∫ 𝑙𝑛𝑥𝑥2 𝑑𝑥

To evaluate the integral ∫ ln(x)/x^2 dx, we can use integration by parts.

Step 1: Let's start by applying the formula for integration by parts, which states ∫ u dv = uv - ∫ v du. In this case, we can choose u = ln(x) and dv = 1/x^2 dx.

Step 2: Next, we need to find du and v. Taking the derivative of u, we have du = (1/x) dx. To find v, we integrate dv. Integrating 1/x^2 dx gives us v = -1/x.

Step 3: Now, we can substitute the values of u, du, v, and dv into the integration by parts formula. This gives us ∫ ln(x)/x^2 dx = ln(x)(-1/x) - ∫ (-1/x)(1/x) dx.

Step 4: Simplifying the expression, we have ∫ ln(x)/x^2 dx = -ln(x)/x + ∫ 1/x^2 dx.

Step 5: The integral ∫ 1/x^2 dx can be evaluated as -1/x. Therefore, our final result is ∫ ln(x)/x^2 dx = -ln(x)/x - (-1/x) + C, where C is the constant of integration.

So, the evaluated integral is ∫ ln(x)/x^2 dx = -ln(x)/x + 1/x + C.