Question

Find the integral ∫13x ln(x) dx using integration by parts with u = 13 ln(x) and dv = x dx. Provide only the function as your answer without including +C.

Answer 1

The integral ∫13x ln(x) dx is solved using integration by parts to yield ⅒13x² ln(x) ⁄ 2 - x³ ⁄ 6 as the function.

Step-by-step explanation:

To find the integral ∫13x ln(x) dx using integration by parts with u = 13 ln(x) and dv = x dx, we need to use the integration by parts formula: ∫ u dv = uv - ∫v du. First, we differentiate u and integrate dv:

• du = ⅓13 ⁄ x dx = 1 dx (because the derivative of ln(x) is 1 ⁄ x)
• v = ⅒x² ⁄ 2 (since the integral of x is x² ⁄ 2)

Now, we plug these into the formula:

13 ln(x) × (⅒x² ⁄ 2) - ∫ (⅒x² ⁄ 2) × 1 dx = ⅒13x² ln(x) ⁄ 2 - ∫ (x² ⁄ 2) dx

Applying the power rule for integrals to ∫ (x² ⁄ 2) dx gives us x³ ⁄ 6. Finally, the function we have is:

⅒13x² ln(x) ⁄ 2 - x³ ⁄ 6