Evaluate the integrals in Exercises 31–78.
35. ∫sec²x e^(tan x)dx

Verified step by step guidance

1. Recognize that the integral involves the function \(e^{\tan x}\) multiplied by \(\sec^2 x\). This suggests a substitution related to the derivative of \(\tan x\).
2. Recall that the derivative of \(\tan x\) is \(\sec^2 x\), which matches the factor multiplying \(e^{\tan x}\) in the integral.
3. Set the substitution \(u = \tan x\), so that \(du = \sec^2 x \, dx\). This allows us to rewrite the integral in terms of \(u\).
4. Rewrite the integral as \(\int e^u \, du\), which is a standard integral involving the exponential function.
5. Integrate \(e^u\) with respect to \(u\) to get \(e^u + C\), then substitute back \(u = \tan x\) to express the answer in terms of \(x\).

Key Concepts

Integration by Substitution

Integration by substitution is a method used to simplify integrals by changing variables. It involves identifying a part of the integrand whose derivative is also present, allowing the integral to be rewritten in terms of a new variable. This technique is especially useful when the integral contains a composite function.

Derivative of the Tangent Function

The derivative of tan(x) is sec²(x). Recognizing this relationship helps in integration problems where sec²(x) appears alongside functions of tan(x). This connection often suggests using tan(x) as a substitution variable to simplify the integral.

Exponential Functions in Integration

Exponential functions like e^(tan x) maintain their form under differentiation and integration, making them straightforward to handle once the substitution is made. Understanding how to integrate expressions involving exponentials combined with trigonometric functions is key to solving such integrals.