Question

Evaluate the integral. $$\int e^{-2 x} \tan \left(e^{-2 x}\right) d x$$

   Evaluate the integral.
$$\int e^{-2 x} \tan \left(e^{-2 x}\right) d x$$

Calculus A New Horizon

Calculus A New Horizon

Howard Anton 6th Edition

Chapter 9, Problem 25

Instant Answer

Solved by Expert Cindy Rodgers

07/07/2021

Step 1

Let $u=e^{-2x}$. Then, $du=-2e^{-2x}dx$ or $dx=-\frac{1}{2}e^{2x}du$. Show more…

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Evaluate the integral. $$\int e^{-2 x} \tan \left(e^{-2 x}\right) d x$$

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Key Concepts

u-Substitution

This technique involves substituting part of the integrand with a new variable (often denoted as u) whose derivative is present in the integrand. It simplifies evaluating integrals involving composite functions by transforming the variable of integration, making the integral more manageable.

Integration of Trigonometric Functions

Integrating trigonometric functions requires knowledge of standard integrals and identities. For example, the integral of the tangent function typically involves using the identity tan(u) = sin(u)/cos(u) and recognizing that its antiderivative is -ln|cos(u)|, which is crucial when the trigonometric function appears within a composite integrand.

Exponential Functions in Integration

Exponential functions often appear in integrals, and their properties allow for effective substitutions. When an exponential function is composed with another function, its derivative is proportional to itself, which simplifies the application of substitution methods and reduces the integral to a standard form.

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