The integral provided is:

$\int_{0}^{1} (1 - x^9) \, dx$

Let's solve this step by step:

Step 1: Break the integral into two parts

$\int_{0}^{1} (1 - x^9) \, dx = \int_{0}^{1} 1 \, dx - \int_{0}^{1} x^9 \, dx$

Step 2: Solve each integral separately

First term:

$\int_{0}^{1} 1 \, dx = [x]_{0}^{1} = 1 - 0 = 1$

Second term:

$\int_{0}^{1} x^9 \, dx = \left[ \frac{x^{10}}{10} \right]_{0}^{1} = \frac{1^{10}}{10} - \frac{0^{10}}{10} = \frac{1}{10} - 0 = \frac{1}{10}$

Step 3: Subtract the results

$\int_{0}^{1} (1 - x^9) \, dx = 1 - \frac{1}{10} = \frac{10}{10} - \frac{1}{10} = \frac{9}{10}$

Final Answer:

$\frac{9}{10}$

Would you like a detailed explanation for any step, or assistance with another problem?

Related Questions:

1. How can you verify this integral using numerical methods?
2. What happens if the upper limit of integration changes, e.g., from 0 to 2?
3. How do we compute definite integrals for more complex expressions like $x^n$?
4. What is the significance of splitting the integral into simpler terms?
5. Can this method generalize for $\int_0^1 (1 - x^n) \, dx$?

Tip:

For polynomials, definite integrals are often simplified by integrating each term separately and applying the Fundamental Theorem of Calculus.