To solve the given problem, we need to evaluate the expression:
\[
7\left(\frac{\int_0^1 x^4(1-x)^4 \, dx}{1+x^2} + \pi\right)
\]
### Step 1: Evaluate the integral \( \int_0^1 x^4(1-x)^4 \, dx \)
Using the binomial expansion, we can express \( (1-x)^4 \):
\[
(1-x)^4 = \sum_{k=0}^{4} \binom{4}{k} (-x)^k = 1 - 4x + 6x^2 - 4x^3 + x^4
\]
Now, we can write:
\[
x^4(1-x)^4 = x^4(1 - 4x + 6x^2 - 4x^3 + x^4) = x^4 - 4x^5 + 6x^6 - 4x^7 + x^8
\]
Thus, we have:
\[
\int_0^1 x^4(1-x)^4 \, dx = \int_0^1 (x^4 - 4x^5 + 6x^6 - 4x^7 + x^8) \, dx
\]
### Step 2: Compute the integral term by term
Calculating each term:
1. \( \int_0^1 x^4 \, dx = \frac{x^5}{5} \bigg|_0^1 = \frac{1}{5} \)
2. \( \int_0^1 4x^5 \, dx = 4 \cdot \frac{x^6}{6} \bigg|_0^1 = \frac{4}{6} = \frac{2}{3} \)
3. \( \int_0^1 6x^6 \, dx = 6 \cdot \frac{x^7}{7} \bigg|_0^1 = \frac{6}{7} \)
4. \( \int_0^1 4x^7 \, dx = 4 \cdot \frac{x^8}{8} \bigg|_0^1 = \frac{4}{8} = \frac{1}{2} \)
5. \( \int_0^1 x^8 \, dx = \frac{x^9}{9} \bigg|_0^1 = \frac{1}{9} \)
Now, substituting back into the integral:
\[
\int_0^1 x^4(1-x)^4 \, dx = \frac{1}{5} - \frac{2}{3} + \frac{6}{7} - \frac{1}{2} + \frac{1}{9}
\]
### Step 3: Find a common denominator and simplify
The common denominator for \( 5, 3, 7, 2, 9 \) is \( 630 \).
Calculating each term:
- \( \frac{1}{5} = \frac{126}{630} \)
- \( \frac{2}{3} = \frac{420}{630} \)
- \( \frac{6}{7} = \frac{540}{630} \)
- \( \frac{1}{2} = \frac{315}{630} \)
- \( \frac{1}{9} = \frac{70}{630} \)
Now substituting these values:
\[
\int_0^1 x^4(1-x)^4 \, dx = \frac{126 - 420 + 540 - 315 + 70}{630} = \frac{1}{630}
\]
### Step 4: Substitute back into the original expression
Now we substitute this back into our original expression:
\[
7\left(\frac{\frac{1}{630}}{1+x^2} + \pi\right)
\]
### Step 5: Simplify the expression
The expression simplifies to:
\[
7\left(\frac{1}{630(1+x^2)} + \pi\right)
\]
### Final Result
Thus, the final answer is:
\[
\frac{7}{630(1+x^2)} + 7\pi
\]
