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

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Recognize that the integral involves the function \(e^{\tan x}\) multiplied by \(\sec^2 x\). This suggests a substitution related to the derivative of \(\tan x\).

Recall that the derivative of \(\tan x\) is \(\sec^2 x\), which matches the factor multiplying \(e^{\tan x}\) in the integral.

Set the substitution \(u = \tan x\), so that \(du = \sec^2 x \, dx\). This allows us to rewrite the integral in terms of \(u\).

Rewrite the integral as \(\int e^u \, du\), which is a standard integral involving the exponential function.

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 \(x\).

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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.

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