$$ \int\left(6x^5+4x^3-6x^2-4x+5\right) $$

The integral you're asking for is:

$$ \int \left(6x^5 + 4x^3 - 6x^2 - 4x + 5 \right) \, dx $$

Let's solve this term by term:

1. $ \int 6x^5 \, dx = \frac{6}{6} x^6 = x^6 $
2. $ \int 4x^3 \, dx = \frac{4}{4} x^4 = x^4 $
3. $ \int -6x^2 \, dx = \frac{-6}{3} x^3 = -2x^3 $
4. $ \int -4x \, dx = \frac{-4}{2} x^2 = -2x^2 $
5. $ \int 5 \, dx = 5x $

Now, putting everything together:

$$ \int \left(6x^5 + 4x^3 - 6x^2 - 4x + 5 \right) \, dx = x^6 + x^4 - 2x^3 - 2x^2 + 5x + C $$

Where $ C $ is the constant of integration.

Would you like further explanation or details on how each term was integrated?

Here are 5 related questions to expand on this topic:
1. How do we handle integration with more complex terms, like trigonometric functions?
2. How can definite integrals be calculated for a polynomial function?
3. What is the relationship between integration and differentiation?
4. Can you explain how to apply substitution in integrals?
5. How can integration be used to find areas under curves?

**Tip:** Always remember to include the constant of integration $ C $ when computing indefinite integrals!

$$ \int \left(6x^5 + 4x^3 - 6x^2 - 4x + 5 \right) $$

Learn how to integrate the polynomial $$ 6x^5 + 4x^3 - 6x^2 - 4x + 5 $$ step-by-step. This solution covers key calculus concepts, including the power rule and constant integration. Suitable for Grades 11-12 and college students.

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