Question

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

Accepted Answer

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 $$\frac{f'(x)}{f(x)} is$$ $$\ln|f(x)| + C. $$For example, the integral $$\int \frac{1}{x} \, dx$$ leads to $$\ln|x| + C$$, since the derivative of $$\ln|x| is$$ $$\frac{1}{x}. To $$find the integrals of $$	an x$$, $$\cot x$$, $$\sec x$$, and $$\csc x$$, rewrite each function in terms of sine and cosine and use substitution where appropriate: - $$	an x = \frac{\sin x}{\cos x}$$, so $$\int 	an x \, dx = \int \frac{\sin x}{\cos x} \, dx$$. - $$\cot x = \frac{\cos x}{\sin x}$$, so $$\int \cot x \, dx = \int \frac{\cos x}{\sin x} \, dx. - $$For $$\sec x$$ and $$\csc x$$, use the trick of multiplying numerator and denominator by expressions involving $$\sec x + 	an x or$$ $$\csc x + \cot x to $$facilitate substitution. Apply substitution for each integral: - For $$\int 	an x \, dx$$, let $$u = \cos x$$, then $$du = -\sin x \, dx. - $$For $$\int \cot x \, dx$$, let $$u = \sin x$$, then $$du = \cos x \, dx. - $$For $$\int \sec x \, dx$$, multiply numerator and denominator by $$\sec x + 	an x$$ and let $$u = \sec x + 	an x. - $$For $$\int \csc x \, dx$$, multiply numerator and denominator by $$\csc x + \cot x$$ and let $$u = \csc x + \cot x. $$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.

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

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Key Concepts

Integrals Leading to Logarithms

Integration of Trigonometric Functions

Use of Substitution in Integration