Evaluate the integrals in Exercises 31–78.
43. ∫tan(ln v)/v dv

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Recognize that the integral is of the form \(\int \frac{\tan(\ln v)}{v} \, dv\). Notice the presence of \(\ln v\) inside the tangent function and the \(\frac{1}{v}\) factor outside, which suggests a substitution involving \(\ln v\).

Let \(u = \ln v\). Then, compute the differential \(du\): since \(u = \ln v\), we have \(du = \frac{1}{v} dv\). This means \(\frac{1}{v} dv = du\).

Rewrite the integral in terms of \(u\): substituting \(u\) and \(du\) gives \(\int \tan(u) \, du\).

Recall the integral formula for \(\int \tan(u) \, du\). The integral of \(\tan(u)\) is \(-\ln|\cos(u)| + C\). Use this to express the integral in terms of \(u\).

Finally, substitute back \(u = \ln v\) to write the answer in terms of the original variable \(v\). The result will be \(-\ln|\cos(\ln v)| + C\).

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Integration by Substitution

Integration by substitution is a method used to simplify integrals by changing variables. It involves identifying a part of the integrand as a new variable, which transforms the integral into a simpler form. This technique is especially useful when the integral contains a composite function, such as tan(ln v).

Properties of Logarithmic Functions

Logarithmic functions, like ln(v), have specific properties that help in integration, such as their derivatives and the chain rule application. Understanding how ln(v) behaves and how its derivative 1/v appears in the integral is crucial for choosing the correct substitution.

Integration of Trigonometric Functions

Integrating trigonometric functions like tan(x) requires knowledge of their standard integrals. For example, ∫tan(x) dx = -ln|cos(x)| + C. Recognizing this allows one to integrate tan(ln v) after substitution, linking trigonometric integration with logarithmic substitution.

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