To solve the integral \( \int x^5 \cos^2(\log(\sqrt{x})) \, dx \), we can follow these steps:
### Step 1: Simplify the Integral
We start by simplifying the cosine squared term using the identity:
\[
\cos^2(\theta) = \frac{1 + \cos(2\theta)}{2}
\]
Applying this to our integral:
\[
\cos^2(\log(\sqrt{x})) = \frac{1 + \cos(2\log(\sqrt{x}))}{2}
\]
Since \( \log(\sqrt{x}) = \frac{1}{2} \log(x) \), we have:
\[
\cos(2\log(\sqrt{x})) = \cos(\log(x))
\]
Thus, the integral becomes:
\[
\int x^5 \cos^2(\log(\sqrt{x})) \, dx = \frac{1}{2} \int x^5 (1 + \cos(\log(x))) \, dx
\]
### Step 2: Separate the Integral
Now, we can separate the integral into two parts:
\[
\frac{1}{2} \left( \int x^5 \, dx + \int x^5 \cos(\log(x)) \, dx \right)
\]
### Step 3: Solve the First Integral
The first integral is straightforward:
\[
\int x^5 \, dx = \frac{x^{6}}{6} + C_1
\]
### Step 4: Solve the Second Integral Using Integration by Parts
Let \( I = \int x^5 \cos(\log(x)) \, dx \). We will use integration by parts where:
- Let \( u = x^5 \) and \( dv = \cos(\log(x)) \, dx \)
Calculating \( du \) and \( v \):
- \( du = 5x^4 \, dx \)
- To find \( v \), we need to integrate \( dv \):
\[
v = \int \cos(\log(x)) \, dx
\]
Using the substitution \( t = \log(x) \) (thus \( dt = \frac{1}{x} \, dx \) or \( dx = e^t \, dt \)):
\[
\int \cos(t) e^t \, dt
\]
Using integration by parts again:
- Let \( a = e^t \) and \( db = \cos(t) \, dt \)
- Then \( da = e^t \, dt \) and \( b = \sin(t) \)
Thus:
\[
\int e^t \cos(t) \, dt = e^t \sin(t) - \int e^t \sin(t) \, dt
\]
This integral can be solved similarly. However, we can denote:
\[
\int e^t \cos(t) \, dt = e^t \sin(t) + e^t \cos(t) + C
\]
Returning to our original variables, we have:
\[
v = x \sin(\log(x)) + x \cos(\log(x))
\]
### Step 5: Substitute Back into Integration by Parts
Now substituting back into the integration by parts formula:
\[
I = x^5 (x \sin(\log(x)) + x \cos(\log(x))) - \int 5x^4 (x \sin(\log(x)) + x \cos(\log(x))) \, dx
\]
### Step 6: Combine Results
Finally, we combine our results:
\[
\int x^5 \cos^2(\log(\sqrt{x})) \, dx = \frac{1}{2} \left( \frac{x^6}{6} + I \right)
\]
### Final Result
After solving for \( I \) and substituting back, we will arrive at the final expression for the integral.
