Integrate this function f (x) = 3x^2 + 6x − 9

To find the integral of the function \( f(x) = 3x^2 + 6x - 9 \), we apply the power rule for integration to each term.

The general rule for integrating a power of \(x\) is:

\[
\int x^n \, dx = \frac{x^{n+1}}{n+1} + C
\]

Now, let's integrate each term:

\[
\int (3x^2 + 6x - 9) \, dx = \int 3x^2 \, dx + \int 6x \, dx + \int -9 \, dx
\]

1. For the first term \( 3x^2 \):

\[
\int 3x^2 \, dx = 3 \cdot \frac{x^{2+1}}{2+1} = 3 \cdot \frac{x^3}{3} = x^3
\]

2. For the second term \( 6x \):

\[
\int 6x \, dx = 6 \cdot \frac{x^{1+1}}{1+1} = 6 \cdot \frac{x^2}{2} = 3x^2
\]

3. For the third term \( -9 \):

\[
\int -9 \, dx = -9x
\]

Putting it all together:

\[
\int (3x^2 + 6x - 9) \, dx = x^3 + 3x^2 - 9x + C
\]

where \(C\) is the constant of integration.

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Would you like further details or explanations? 

Here are 5 related questions:

1. How do you find definite integrals?
2. What is the geometric interpretation of an integral?
3. How do you apply integration to physics problems?
4. How can you integrate a function with trigonometric terms?
5. What happens if the function has a fractional exponent?

**Tip:** Always remember to add the constant of integration \(C\) when performing indefinite integrals!

Integrate this function f(x) = 3x^2 + 6x − 9

Learn how to integrate the polynomial function f(x) = 3x^2 + 6x - 9 using the power rule for integration. This detailed guide walks through each step of the process, making it easy for students in Grades 11-12 or introductory college levels to understand. Includes formulas and key integration concepts.

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