Question

The following indefinite integral can be evaluated using integration by parts with the given choices of u and dv. ∫x2ln(x)dx;u=ln(x),dv=x2dx Determine du and v. du= −v= −v Evaluate the integral.

Answer 1

To evaluate the indefinite integral ∫x^2ln(x)dx using integration by parts, we let u = ln(x) and dv = x^2dx. We find du = (1/x)dx and v = (1/3)x^3. Plugging these values into the integration by parts formula, we can simplify the integral and find the final result.

Step-by-step explanation:

To evaluate the indefinite integral ∫x^2ln(x)dx using integration by parts, we need to determine u and dv. In this case, u = ln(x) and dv = x^2dx. To find du, we take the derivative of u with respect to x, which is du = (1/x)dx. To find v, we integrate dv with respect to x, which is v = (1/3)x^3.

Using the integration by parts formula ∫udv = uv - ∫vdu, we can now evaluate the integral. Plugging the values we found for u, dv, du, and v into the formula, we have:

∫x^2ln(x)dx = (ln(x) * (1/3)x^3) - ∫(1/3)x^3 * (1/x)dx = (1/3)x^3ln(x) - (1/3)∫x^2dx

Simplifying further, we have ∫x^2dx = (1/3)x^3, which gives:

∫x^2ln(x)dx = (1/3)x^3ln(x) - (1/3)(1/3)x^3

So, the final result is ∫x^2ln(x)dx = (1/3)x^3ln(x) - (1/3)(1/3)x^3 + C, where C is the constant of integration.