Question Statement

Evaluate the indefinite integral:

$\int \left(x^2-3x+9\right)dx$

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Background and Explanation

This problem involves integrating a polynomial function, which requires applying the power rule for integration and the linearity property of integrals. These fundamental techniques allow you to integrate term by term.

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Solution

Let $I$ represent the integral:

$I=\int \left(x^2-3x+9\right)dx$

Step 1: Apply the linearity property of integration to split the integral into separate terms and factor out constants:

$I=\int x^{2}dx-3\int xdx+9\int 1dx$

Reasoning: The integral of a sum equals the sum of the integrals, and constants can be factored out.

Step 2: Apply the power rule $\int x^{n}dx=\frac{x^{n+1}}{n+1}$ to each term. For the first term, $n=2$; for the second term, $n=1$; and for the constant, we use $\int 1dx=x$:

$I=\frac{x^{2+1}}{2+1}-3\cdot \frac{x^{1+1}}{1+1}+9x+c$

Note: Here we explicitly show the power rule application with $n+1$ in the exponents. The constant of integration $c$ is added because this is an indefinite integral.

Step 3: Simplify the exponents and coefficients:

$I=\frac{x^3}{3}-3\cdot \frac{x^2}{2}+9x+c$

Step 4: Final simplification:

$I=\frac{x^3}{3}-\frac{3x^2}{2}+9x+c$

Therefore, the solution is:

$\int \left(x^2-3x+9\right)dx=\frac{x^3}{3}-\frac{3x^2}{2}+9x+c$

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Key Formulas or Methods Used

• Linearity of Integration: $\int [af(x)+bg(x)]dx=a\int f(x)dx+b\int g(x)dx$
• Power Rule: $\int x^{n}dx=\frac{x^{n+1}}{n+1}+c$ (for $n\ne -1$)
• Constant Rule: $\int 1dx=x+c$ or $\int kdx=kx+c$ (where $k$ is a constant)

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Summary of Steps

1. Set up the integral and denote it as $I$
2. Apply linearity to separate the integral into three individual terms
3. Integrate each term using the power rule $\int x^{n}dx=\frac{x^{n+1}}{n+1}$, showing the explicit substitution of $n=2$ and $n=1$
4. Add the constant of integration $c$ at the end
5. Simplify the resulting algebraic expression