We need to evaluate the definite integral $K = \int_0^1 (x^2 + 1)^7 x \, dx$.

1. Problem Description

We need to evaluate the definite integral $K = \int_0^1 (x^2 + 1)^7 x \, dx$.

2. Solution Steps

We can solve this integral using substitution.

Let $u = x^2 + 1$.

Then, $\frac{du}{dx} = 2x$, so $du = 2x \, dx$.

Thus, $x \, dx = \frac{1}{2} du$.

When $x = 0$, $u = 0^2 + 1 = 1$.

When $x = 1$, $u = 1^2 + 1 = 2$.

Now we can rewrite the integral in terms of $u$:

$K = \int_1^2 u^7 \cdot \frac{1}{2} du = \frac{1}{2} \int_1^2 u^7 du$.

We can evaluate the integral of $u^7$ using the power rule for integration:

$\int u^n du = \frac{u^{n+1}}{n+1} + C$

So, $\int u^7 du = \frac{u^8}{8} + C$.

Therefore, $K = \frac{1}{2} \left[ \frac{u^8}{8} \right]_1^2 = \frac{1}{2} \left( \frac{2^8}{8} - \frac{1^8}{8} \right) = \frac{1}{2} \left( \frac{256}{8} - \frac{1}{8} \right) = \frac{1}{2} \left( \frac{255}{8} \right) = \frac{255}{16}$.

3. Final Answer

$K = \frac{255}{16}$