Is $\int \frac{e^x}{e^x+5} dx$ a candidate for logarithmic integration?

Yes, because the derivative of $e^x+5$ is exactly $e^x$, which is the numerator. The result is $\ln|e^x+5| + c$.

What is the general formula for $\int \frac{f'(x)}{f(x)} dx$?

The result is $\ln|f(x)| + c$. This applies whenever the numerator is the exact derivative of the denominator.

Why must we use the modulus sign in the result $\ln|f(x)|$?

The natural logarithm function is only defined for positive arguments. The modulus ensures that even if $f(x)$ is negative, the integral remains valid.

How does integrating $\frac{1}{x}$ differ from integrating $\frac{1}{x^2}$?

$\frac{1}{x}$ follows the $f'(x)/f(x)$ pattern resulting in $\ln|x|$, whereas $\frac{1}{x^2}$ is treated as $x^{-2}$ and follows the power rule, resulting in $-x^{-1}$.

When should you use 'Adjust and Compensate' in fractional integration?

Use it when the numerator is a constant multiple of the denominator's derivative. You multiply the numerator to make it the exact derivative and divide the outside of the integral by that same constant.

What is the result of $\int \tan(x) dx$ using the $f'(x)/f(x)$ rule?

Rewrite $\tan(x)$ as $\frac{\sin x}{\cos x}$. Since the derivative of $\cos x$ is $-\sin x$, the integral becomes $-\ln|\cos x| + c$, which is also written as $\ln|\sec x| + c$.

What error occurs if you integrate $\int \frac{3}{2x+1} dx$ as $3\ln|2x+1|$?

The derivative of $2x+1$ is $2$, not $1$. The correct integral requires a compensation factor of $1/2$, resulting in $\frac{3}{2}\ln|2x+1| + c$.

Can the $f'(x)/f(x)$ rule be used if the denominator is $(x^2+1)^2$?

No. The rule only applies when the entire denominator is to the power of 1. If it is squared, you must use the power rule (reverse chain rule) or substitution.

What is the first step when faced with a complex rational function integral?

Differentiate the denominator. If the result is a scalar multiple of the numerator, use the logarithmic integration rule.

How do you handle a negative sign when the numerator is $-\sin(x)$ and the denominator is $\cos(x)$?

Since $-\sin(x)$ is the exact derivative of $\cos(x)$, the integral is simply $\ln|\cos(x)| + c$ without needing further adjustment.

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Integrating f'(x)/f(x)

Summary

Integrating functions of the form $f'(x)/f(x)$ is a specialized application of the reverse chain rule that results in a natural logarithmic function. This technique is essential for solving integrals where the numerator is the derivative (or a scalar multiple of the derivative) of the denominator.

1. Definition & Core Concepts

Diagram showing the fractional form f'(x) over f(x) leading to the natural log result.

2. Underlying Principles

3. Methods & Techniques: Adjust and Compensate

4. Key Distinctions

5. Exam Strategy & Tips

6. Common Pitfalls & Misconceptions

Ignoring the Power: A common mistake is trying to use $\ln$ for $\int \frac{1}{x^2} dx$ . This is incorrect because the denominator's power is 2; the power rule must be used ( $x^{-2}$ ).
Incorrect Compensation: Students often multiply the inside by a constant but forget to divide the outside by the same constant, leading to an answer that is off by a factor.
Forgetting the Modulus: Omitting the absolute value signs can lead to undefined results if the function $f(x)$ takes negative values within the limits of integration.