Power Rule for Integration worksheet with ten problems for finding indefinite integrals.

Let's solve each of these integrals using the Power Rule for Integration.

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🔷 Power Rule for Integration:

For any real number $ n \ne -1 $:

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

Also, remember:
- The integral of a sum is the sum of the integrals.
- Constants can be factored out.
- For negative exponents: $ \frac{1}{x^n} = x^{-n} $

We'll apply this rule to each problem.

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1) $ \int (2x - 1)\,dx $

Break it into two parts:

$$
= \int 2x\,dx - \int 1\,dx
= 2 \int x\,dx - \int dx
= 2 \cdot \frac{x^2}{2} - x + C
= x^2 - x + C
$$

✔ Answer: $ x^2 - x + C $

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2) $ \int (-4x^3 + 36x^2 - 72x)\,dx $

Integrate term by term:

$$
= -4 \int x^3\,dx + 36 \int x^2\,dx - 72 \int x\,dx
= -4 \cdot \frac{x^4}{4} + 36 \cdot \frac{x^3}{3} - 72 \cdot \frac{x^2}{2}
= -x^4 + 12x^3 - 36x^2 + C
$$

✔ Answer: $ -x^4 + 12x^3 - 36x^2 + C $

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3) $ \int (2x + 3)\,dx $

$$
= 2 \int x\,dx + \int 3\,dx
= 2 \cdot \frac{x^2}{2} + 3x + C
= x^2 + 3x + C
$$

✔ Answer: $ x^2 + 3x + C $

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4) $ \int \left( \frac{-1}{x^2} \right) dx $

Rewrite: $ \frac{-1}{x^2} = -x^{-2} $

$$
= \int -x^{-2}\,dx = - \int x^{-2}\,dx
= - \cdot \frac{x^{-1}}{-1} + C = x^{-1} + C = \frac{1}{x} + C
$$

✔ Answer: $ \frac{1}{x} + C $

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5) $ \int \left( \frac{-57}{x^4} \right) dx $

$ \frac{-57}{x^4} = -57x^{-4} $

$$
= -57 \int x^{-4}\,dx = -57 \cdot \frac{x^{-3}}{-3} + C
= 19x^{-3} + C = \frac{19}{x^3} + C
$$

✔ Answer: $ \frac{19}{x^3} + C $

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6) $ \int \left( \frac{-28}{x^2} \right) dx $

$ = -28 \int x^{-2}\,dx = -28 \cdot \frac{x^{-1}}{-1} + C = 28x^{-1} + C = \frac{28}{x} + C $

✔ Answer: $ \frac{28}{x} + C $

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7) $ \int \left( \frac{12}{169x^{13}} \right) dx $

Simplify: $ \frac{12}{169} \cdot x^{-13} $

$$
= \frac{12}{169} \int x^{-13}\,dx = \frac{12}{169} \cdot \frac{x^{-12}}{-12} + C
= -\frac{1}{169} x^{-12} + C = -\frac{1}{169x^{12}} + C
$$

✔ Answer: $ -\frac{1}{169x^{12}} + C $

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8) $ \int \left( \frac{-3}{11x^{11}} \right) dx $

$ = -\frac{3}{11} \int x^{-11}\,dx = -\frac{3}{11} \cdot \frac{x^{-10}}{-10} + C = \frac{3}{110} x^{-10} + C = \frac{3}{110x^{10}} + C $

✔ Answer: $ \frac{3}{110x^{10}} + C $

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9) $ \int (2x + 5)\,dx $

$$
= 2 \int x\,dx + \int 5\,dx = 2 \cdot \frac{x^2}{2} + 5x + C = x^2 + 5x + C
$$

✔ Answer: $ x^2 + 5x + C $

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10) $ \int \left( \frac{-10}{133x^9} \right) dx $

$ = -\frac{10}{133} \int x^{-9}\,dx = -\frac{10}{133} \cdot \frac{x^{-8}}{-8} + C = \frac{10}{133 \cdot 8} x^{-8} + C $

$$
= \frac{10}{1064} x^{-8} + C = \frac{5}{532} x^{-8} + C = \frac{5}{532x^8} + C
$$

✔ Answer: $ \frac{5}{532x^8} + C $

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✔ Final Answers Summary:

| Problem | Answer |
|--------|--------|
| 1) | $ x^2 - x + C $ |
| 2) | $ -x^4 + 12x^3 - 36x^2 + C $ |
| 3) | $ x^2 + 3x + C $ |
| 4) | $ \frac{1}{x} + C $ |
| 5) | $ \frac{19}{x^3} + C $ |
| 6) | $ \frac{28}{x} + C $ |
| 7) | $ -\frac{1}{169x^{12}} + C $ |
| 8) | $ \frac{3}{110x^{10}} + C $ |
| 9) | $ x^2 + 5x + C $ |
| 10) | $ \frac{5}{532x^8} + C $ |

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