Certainly! Let's work through the integral \(\int x^7 \ln (x^5) \, dx\) step by step.
### Step 1: Simplify the Integrand
First, notice that \(\ln (x^5)\) can be simplified using the properties of logarithms:
[tex]\[
\ln (x^5) = 5 \ln (x)
\][/tex]
Thus, the integral becomes:
[tex]\[
\int x^7 \ln (x^5) \, dx = \int x^7 \cdot 5 \ln (x) \, dx = 5 \int x^7 \ln (x) \, dx
\][/tex]
### Step 2: Use Integration by Parts
To integrate \(5 \int x^7 \ln (x) \, dx\), we'll use the method of integration by parts, which states:
[tex]\[
\int u \, dv = uv - \int v \, du
\][/tex]
Let's choose \(u\) and \(dv\):
[tex]\[
u = \ln (x) \quad \Rightarrow \quad du = \frac{1}{x} \, dx
\][/tex]
[tex]\[
dv = x^7 \, dx \quad \Rightarrow \quad v = \frac{x^8}{8}
\][/tex]
### Step 3: Apply Integration by Parts
Using integration by parts, we get:
[tex]\[
\int x^7 \ln (x) \, dx = \left. \ln (x) \cdot \frac{x^8}{8} \right| - \int \frac{x^8}{8} \cdot \frac{1}{x} \, dx
\][/tex]
Simplifying the integral on the right-hand side:
[tex]\[
= \frac{x^8 \ln (x)}{8} - \int \frac{x^8}{8} \cdot \frac{1}{x} \, dx = \frac{x^8 \ln (x)}{8} - \int \frac{x^7}{8} \, dx
\][/tex]
### Step 4: Simplify the Remaining Integral
Now, let's integrate \(\frac{x^7}{8}\):
[tex]\[
\int \frac{x^7}{8} \, dx = \frac{1}{8} \int x^7 \, dx = \frac{1}{8} \cdot \frac{x^8}{8} = \frac{x^8}{64}
\][/tex]
### Step 5: Combine the Results
Putting everything together, we have:
[tex]\[
\int x^7 \ln (x) \, dx = \frac{x^8 \ln (x)}{8} - \frac{x^8}{64}
\][/tex]
### Step 6: Multiply by the Constant
Don't forget the constant \(5\) we factored out earlier:
[tex]\[
5 \int x^7 \ln (x) \, dx = 5 \left( \frac{x^8 \ln (x)}{8} - \frac{x^8}{64} \right)
\][/tex]
Simplify the expression:
[tex]\[
5 \left( \frac{x^8 \ln (x)}{8} - \frac{x^8}{64} \right) = \frac{5x^8 \ln (x)}{8} - \frac{5x^8}{64}
\][/tex]
### Conclusion
So, the final result is:
[tex]\[
\int x^7 \ln (x^5) \, dx = \frac{5 x^8 \ln (x)}{8} - \frac{5 x^8}{64}
\][/tex]
This concludes our detailed, step-by-step solution.
