Calculus Worksheets | Indefinite Integration Worksheets - Free Printable

Certainly! Let’s solve each of the 10 indefinite integrals using the Power Rule for Integration, which states:

> For any real number \( n \neq -1 \),
> \[
\int x^n \, dx = \frac{x^{n+1}}{n+1} + C
\]
> Also, constants can be factored out: \( \int k \cdot f(x) \, dx = k \int f(x) \, dx \)

We’ll go one by one.

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

Split the integral:
\[
= \int 2x \, dx - \int 1 \, dx = 2 \int x^1 \, dx - \int x^0 \, dx
\]

Apply power rule:
\[
= 2 \cdot \frac{x^{2}}{2} - \frac{x^{1}}{1} + C = x^2 - x + C
\]

✔ Answer: \( x^2 - x + C \)

---

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

Integrate term by term:
\[
= -4 \int x^3 \, dx + 36 \int x^2 \, dx - 72 \int x \, dx
\]

Apply power rule:
\[
= -4 \cdot \frac{x^4}{4} + 36 \cdot \frac{x^3}{3} - 72 \cdot \frac{x^2}{2} + C
\]

Simplify:
\[
= -x^4 + 12x^3 - 36x^2 + C
\]

✔ Answer: \( -x^4 + 12x^3 - 36x^2 + C \)

---

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

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

✔ Answer: \( x^2 + 3x + C \)

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

Rewrite as power:
\[
= \int -x^{-2} \, dx = - \int x^{-2} \, dx
\]

Apply power rule (\( n = -2 \), so \( n+1 = -1 \)):
\[
= - \cdot \frac{x^{-1}}{-1} + C = \frac{1}{x} + C
\]

✔ Answer: \( \frac{1}{x} + C \)

---

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

Rewrite:
\[
= \int -57 x^{-4} \, dx = -57 \int x^{-4} \, dx
\]

Apply power rule:
\[
= -57 \cdot \frac{x^{-3}}{-3} + C = 19 x^{-3} + C = \frac{19}{x^3} + C
\]

✔ Answer: \( \frac{19}{x^3} + C \)

---

6) \( \int \left( \frac{-28}{x^9} \right) dx \)

Rewrite:
\[
= \int -28 x^{-9} \, dx = -28 \int x^{-9} \, dx
\]

Apply power rule:
\[
= -28 \cdot \frac{x^{-8}}{-8} + C = \frac{28}{8} x^{-8} + C = \frac{7}{2} x^{-8} + C = \frac{7}{2x^8} + C
\]

✔ Answer: \( \frac{7}{2x^8} + C \)

---

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

Factor constant:
\[
= \frac{12}{169} \int x^{-13} \, dx
\]

Apply power rule:
\[
= \frac{12}{169} \cdot \frac{x^{-12}}{-12} + C = -\frac{1}{169} x^{-12} + C = -\frac{1}{169 x^{12}} + C
\]

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

---

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

Factor constant:
\[
= -\frac{3}{11} \int x^{-11} \, dx
\]

Apply power rule:
\[
= -\frac{3}{11} \cdot \frac{x^{-10}}{-10} + C = \frac{3}{110} x^{-10} + C = \frac{3}{110 x^{10}} + C
\]

✔ Answer: \( \frac{3}{110 x^{10}} + C \)

---

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

Same as problem 3:
\[
= \int 2x \, dx + \int 5 \, dx = x^2 + 5x + C
\]

✔ Answer: \( x^2 + 5x + C \)

---

10) \( \int \left( \frac{-10}{133 x^7} \right) dx \)

Factor constant:
\[
= -\frac{10}{133} \int x^{-7} \, dx
\]

Apply power rule:
\[
= -\frac{10}{133} \cdot \frac{x^{-6}}{-6} + C = \frac{10}{798} x^{-6} + C = \frac{5}{399} x^{-6} + C = \frac{5}{399 x^6} + C
\]

✔ Answer: \( \frac{5}{399 x^6} + C \)

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## ✔ Final Answers Summary:

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. \( \frac{7}{2x^8} + C \)
7. \( -\frac{1}{169 x^{12}} + C \)
8. \( \frac{3}{110 x^{10}} + C \)
9. \( x^2 + 5x + C \)
10. \( \frac{5}{399 x^6} + C \)

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