Question

Evaluate the integral ∫ t⁴ ln(t) dt. (Use c for the constant of integration.)

Answer 1

To evaluate the integral of t^4 ln(t) dt, we use integration by parts, with u=ln(t) and dv=t^4 dt, and find that the final answer is (t^5)/5 (ln(t) - 1/5) + c.

Step-by-step explanation:

To evaluate the integral ∫ t^4 ∙ ln(t) dt, we can use integration by parts, which states that ∫ u dv = uv - ∫ v du. We let u = ln(t) and dv = t^4 dt. Then we find that du = (1/t) dt and v = (t^5)/5. Applying integration by parts gives us:

• ∫ t^4 ∙ ln(t) dt = (ln(t) ∙(t^5)/5) - ∫ ((t^5)/5) ∙ (1/t) dt
• = (ln(t) ∙(t^5)/5) - (1/5) ∫ t^4 dt
• = (ln(t) ∙(t^5)/5) - (1/5) ∙ (t^5)/5 + c
• = (t^5)/5 ∙ (ln(t) - 1/5) + c

So the final answer for the integral is (t^5)/5 ∙ (ln(t) - 1/5) + c.