Questions: Use integration by parts to evaluate the integral: [ int fracln (t)t^7 d t= ]

$Use integration by parts to evaluate the integral: [ int fracln (t)t^7 d t= ]$

Transcript text: Use integration by parts to evaluate the integral: \[ \int \frac{\ln (t)}{t^{7}} d t= \]

Solution

Solution Steps

To solve the integral $\int \frac{\ln (t)}{t^{7}} d t$ using integration by parts, we need to identify parts of the integrand to differentiate and integrate. We can let $u = \ln(t)$ and $dv = \frac{1}{t^7} dt$. Then, we differentiate $u$ to find $du$ and integrate $dv$ to find $v$. Finally, we apply the integration by parts formula: $\int u \, dv = uv - \int v \, du$.

Step 1: Set Up the Integral

We start with the integral we want to evaluate: \[ I = \int \frac{\ln(t)}{t^7} \, dt \]

Step 2: Apply Integration by Parts

Using integration by parts, we let: \[ u = \ln(t) \quad \text{and} \quad dv = \frac{1}{t^7} \, dt \] Then, we differentiate and integrate to find: \[ du = \frac{1}{t} \, dt \quad \text{and} \quad v = -\frac{1}{6t^6} \]

Step 3: Compute the Integral

Applying the integration by parts formula: \[ I = uv - \int v \, du \] Substituting in our values: \[ I = \left(-\frac{\ln(t)}{6t^6}\right) - \int \left(-\frac{1}{6t^6}\right) \left(\frac{1}{t}\right) dt \] This simplifies to: \[ I = -\frac{\ln(t)}{6t^6} + \frac{1}{6} \int \frac{1}{t^7} \, dt \]

Step 4: Evaluate the Remaining Integral

The remaining integral is: \[ \int \frac{1}{t^7} \, dt = -\frac{1}{6t^6} \] Thus, we have: \[ I = -\frac{\ln(t)}{6t^6} + \frac{1}{6} \left(-\frac{1}{6t^6}\right) \] This results in: \[ I = -\frac{\ln(t)}{6t^6} - \frac{1}{36t^6} \]