Questions: Evaluate the integral. (Use C for the constant of integration.) ∫ t^4 ln(t) dt

Solution Steps

Let \( u = \ln(t) \) and \( dv = t^4 \, dt \).

Differentiate \( u \) to find \( du \): \[ du = \frac{1}{t} \, dt \] Integrate \( dv \) to find \( v \): \[ v = \int t^4 \, dt = \frac{t^5}{5} \]

Using the integration by parts formula \( \int u \, dv = uv - \int v \, du \), we have: \[ \int t^4 \ln(t) \, dt = \ln(t) \cdot \frac{t^5}{5} - \int \frac{t^5}{5} \cdot \frac{1}{t} \, dt \]

The integral simplifies to: \[ \int t^4 \ln(t) \, dt = \frac{t^5 \ln(t)}{5} - \frac{1}{5} \int t^4 \, dt \]

Now, evaluate the remaining integral: \[ \int t^4 \, dt = \frac{t^5}{5} \] Substituting this back, we get: \[ \int t^4 \ln(t) \, dt = \frac{t^5 \ln(t)}{5} - \frac{1}{5} \cdot \frac{t^5}{5} \]

Combining the results gives: \[ \int t^4 \ln(t) \, dt = \frac{t^5 \ln(t)}{5} - \frac{t^5}{25} + C \] where \( C \) is the constant of integration.

Final Answer