7. What integrals lead to logarithms? Give examples. What are the integrals of tan x, cot x, sec x, and csc x?

Verified step by step guidance

1. Recognize that integrals leading to logarithms typically involve functions whose derivatives are rational functions of the variable, especially those of the form \(\frac{f'(x)}{f(x)}\), because the integral of \(\frac{f'(x)}{f(x)}\) is \(\ln|f(x)| + C\).
2. For example, the integral \(\int \frac{1}{x} \, dx\) leads to \(\ln|x| + C\), since the derivative of \(\ln|x|\) is \(\frac{1}{x}\).
3. To find the integrals of \(\tan x\), \(\cot x\), \(\sec x\), and \(\csc x\), rewrite each function in terms of sine and cosine and use substitution where appropriate:
4. - \(\tan x = \frac{\sin x}{\cos x}\), so \(\int \tan x \, dx = \int \frac{\sin x}{\cos x} \, dx\).
5. - \(\cot x = \frac{\cos x}{\sin x}\), so \(\int \cot x \, dx = \int \frac{\cos x}{\sin x} \, dx\).
6. - For \(\sec x\) and \(\csc x\), use the trick of multiplying numerator and denominator by expressions involving \(\sec x + \tan x\) or \(\csc x + \cot x\) to facilitate substitution.
7. Apply substitution for each integral:
8. - For \(\int \tan x \, dx\), let \(u = \cos x\), then \(du = -\sin x \, dx\).
9. - For \(\int \cot x \, dx\), let \(u = \sin x\), then \(du = \cos x \, dx\).
10. - For \(\int \sec x \, dx\), multiply numerator and denominator by \(\sec x + \tan x\) and let \(u = \sec x + \tan x\).
11. - For \(\int \csc x \, dx\), multiply numerator and denominator by \(\csc x + \cot x\) and let \(u = \csc x + \cot x\).
12. After substitution, each integral simplifies to a form \(\int \frac{1}{u} \, du\), which integrates to \(\ln|u| + C\), giving the logarithmic form of the integral.

Key Concepts

Integrals Leading to Logarithms

Integrals that involve functions of the form f'(x)/f(x) often result in logarithmic expressions, specifically the natural logarithm ln|f(x)|. This occurs because the derivative of ln|f(x)| is f'(x)/f(x), making such integrals fundamental in calculus.

Integration of Trigonometric Functions

Integrating trigonometric functions like tan x, cot x, sec x, and csc x requires rewriting them in terms of sine and cosine or using substitution techniques. These integrals often lead to logarithmic results due to their derivative relationships with sine and cosine.

Use of Substitution in Integration

Substitution is a method to simplify integrals by changing variables, especially useful when the integrand contains a function and its derivative. For example, integrating tan x involves rewriting it as sin x/cos x and substituting u = cos x, which leads to a logarithmic integral.