To find the antiderivative for each function when the constant $C = 0$, we proceed by integrating each function individually.

Solution

Part (a): $f(x) = \frac{1}{x}$

The antiderivative of $\frac{1}{x}$ with respect to $x$ is: $\int \frac{1}{x} \, dx = \ln |x|$ Since $C = 0$, the antiderivative is: $F(x) = \ln |x|$

Part (b): $g(x) = \frac{5}{x}$

The antiderivative of $\frac{5}{x}$ with respect to $x$ is: $\int \frac{5}{x} \, dx = 5 \ln |x|$ With $C = 0$, the antiderivative is: $G(x) = 5 \ln |x|$

Part (c): $h(x) = 4 - \frac{5}{x}$

The antiderivative of $4 - \frac{5}{x}$ with respect to $x$ is: $\int \left( 4 - \frac{5}{x} \right) dx = 4x - 5 \ln |x|$ Again, with $C = 0$, the antiderivative is: $H(x) = 4x - 5 \ln |x|$

Summary of Results

• (a) $F(x) = \ln |x|$
• (b) $G(x) = 5 \ln |x|$
• (c) $H(x) = 4x - 5 \ln |x|$

Would you like further details or have any questions?

Related Questions

1. How would the antiderivative change if $C \neq 0$?
2. What are the steps to find the derivative of $F(x) = \ln |x|$ to verify it?
3. How does the presence of $\ln |x|$ relate to improper integrals?
4. How would you approach finding the derivative of $4x - 5 \ln |x|$?
5. Why do we use absolute value signs in the logarithm function in these cases?

Tip

Remember, when integrating functions with $\frac{1}{x}$, always consider $\ln |x|$, as the absolute value is crucial for functions that can take both positive and negative values of $x$ (excluding zero).