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

검증된 단계별 안내

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 \(-\ln|\cos(\ln v)| + C\).

비슷한 문제에 대한 검증된 영상 답변:

이 영상 해법은 위 문제에 도움이 된다고 튜터들이 추천한 것입니다.

영상 길이:

도움이 되었나요?

주요 개념

질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.

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.

Evaluate the integrals in Exercises 31–78.43. ∫tan(ln v)/v dv