Logarithmic and Exponential Rules: Indefinite Integral Practice Worksheet

Let's solve each of the indefinite integrals listed in the worksheet. We'll use logarithmic and exponential integration rules, along with basic properties of integrals.

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Key Integration Rules:

1. $\int a \cdot f(x)\,dx = a \int f(x)\,dx$ (Constant multiple rule)
2. $\int x^n\,dx = \frac{x^{n+1}}{n+1} + C$, for $n \ne -1$
3. $\int \frac{1}{x}\,dx = \ln|x| + C$
4. $\int e^x\,dx = e^x + C$
5. $\int a^x\,dx = \frac{a^x}{\ln a} + C$, for $a > 0, a \ne 1$
6. $\int \frac{f'(x)}{f(x)}\,dx = \ln|f(x)| + C$

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Now let’s go through each problem:

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1) $\int (9x^{-1})\,dx$

We can factor out the constant:
$$
= 9 \int x^{-1}\,dx = 9 \ln|x| + C
$$

✔ Answer: $9\ln|x| + C$

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2) $\int \left(\frac{7}{x}\right)dx$

This is:
$$
= 7 \int \frac{1}{x}\,dx = 7\ln|x| + C
$$

✔ Answer: $7\ln|x| + C$

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3) $\int (12e^x)\,dx$

Factor out the constant:
$$
= 12 \int e^x\,dx = 12e^x + C
$$

✔ Answer: $12e^x + C$

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4) $\int (4 \cdot 16^x)\,dx$

Factor out the constant:
$$
= 4 \int 16^x\,dx
$$

Use the rule: $\int a^x\,dx = \frac{a^x}{\ln a} + C$

So:
$$
= 4 \cdot \frac{16^x}{\ln 16} + C
$$

Note: $\ln 16 = \ln(2^4) = 4\ln 2$, so we could write it as:
$$
= \frac{4}{4\ln 2} \cdot 16^x + C = \frac{1}{\ln 2} \cdot 16^x + C
$$

But unless simplified, keep it as:
$$
= \frac{4 \cdot 16^x}{\ln 16} + C
$$

✔ Answer: $\frac{4 \cdot 16^x}{\ln 16} + C$

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5) $\int \left(\frac{8}{x}\right)dx$

$$
= 8 \int \frac{1}{x}\,dx = 8\ln|x| + C
$$

✔ Answer: $8\ln|x| + C$

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6) $\int (6e^x)\,dx$

$$
= 6 \int e^x\,dx = 6e^x + C
$$

✔ Answer: $6e^x + C$

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7) $\int (7 \cdot 3^x)\,dx$

Factor out the constant:
$$
= 7 \int 3^x\,dx = 7 \cdot \frac{3^x}{\ln 3} + C
$$

✔ Answer: $\frac{7 \cdot 3^x}{\ln 3} + C$

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8) $\int (5x^{-1})\,dx$

$$
= 5 \int x^{-1}\,dx = 5\ln|x| + C
$$

✔ Answer: $5\ln|x| + C$

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9) $\int (-8e^x)\,dx$

$$
= -8 \int e^x\,dx = -8e^x + C
$$

✔ Answer: $-8e^x + C$

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10) $\int (-9 \cdot 13^x)\,dx$

Factor out the constant:
$$
= -9 \int 13^x\,dx = -9 \cdot \frac{13^x}{\ln 13} + C
$$

✔ Answer: $-\frac{9 \cdot 13^x}{\ln 13} + C$

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✔ Final Answers Summary:

1) $9\ln|x| + C$
2) $7\ln|x| + C$
3) $12e^x + C$
4) $\frac{4 \cdot 16^x}{\ln 16} + C$
5) $8\ln|x| + C$
6) $6e^x + C$
7) $\frac{7 \cdot 3^x}{\ln 3} + C$
8) $5\ln|x| + C$
9) $-8e^x + C$
10) $-\frac{9 \cdot 13^x}{\ln 13} + C$

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Let me know if you'd like these written neatly in LaTeX or explained further!