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6) $\int_1^2 (10x^3 + 4x^2 + 6) \, dx$

We can find this integral by first finding the indefinite integral of the function and then evaluating the result at the upper and lower limits.

Let me calculate this for you. sympy import symbols, integrate

Define the variable and function

x = symbols('x') f = 10x**3 + 4x**2 + 6

Calculate the definite integral from 1 to 2

integral_6 = integrate(f, (x, 1, 2)) integral_6The value of the definite integral $\int_1^2 (10x^3 + 4x^2 + 6) \, dx$ is $\frac{317}{6}$.

Would you like to continue solving the rest of the integrals or need any additional details on this one? Here are a few related questions that might help expand your understanding:

1. How do you evaluate a definite integral step-by-step?
2. What is the Fundamental Theorem of Calculus and how does it apply to definite integrals?
3. How do you find the indefinite integral of a polynomial function?
4. What are the differences between definite and indefinite integrals?
5. How would you evaluate integrals with more complex functions, such as trigonometric or exponential?

Tip: Always check the degree of the polynomial when solving definite integrals — the steps for integration vary slightly for higher-degree polynomials!