Evaluate the integrals in Exercises 41–60.
55. ∫(from -π/4 to π/4)cosh(tanθ)sec²θ dθ

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Identify the integral to evaluate: \(\int_{-\frac{\pi}{4}}^{\frac{\pi}{4}} \cosh(\tan \theta) \sec^{2} \theta \, d\theta\).

Recognize that the integrand contains \(\cosh(\tan \theta)\) and \(\sec^{2} \theta\), and note that the derivative of \(\tan \theta\) is \(\sec^{2} \theta\).

Use the substitution method by letting \(u = \tan \theta\). Then, compute \(du = \sec^{2} \theta \, d\theta\), which matches part of the integrand.

Change the limits of integration according to the substitution: when \(\theta = -\frac{\pi}{4}\), \(u = \tan(-\frac{\pi}{4}) = -1\); when \(\theta = \frac{\pi}{4}\), \(u = \tan(\frac{\pi}{4}) = 1\).

Rewrite the integral in terms of \(u\) as \(\int_{-1}^{1} \cosh(u) \, du\), which is easier to integrate.

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Hyperbolic Cosine Function (cosh)

The hyperbolic cosine function, cosh(x), is defined as (e^x + e^(-x))/2. It is an even function and appears frequently in calculus involving hyperbolic functions. Understanding its properties helps in evaluating integrals where cosh is composed with other functions.

Substitution Method in Integration

The substitution method involves changing variables to simplify an integral. By identifying a part of the integrand as a function whose derivative also appears, we can rewrite the integral in terms of a new variable, making it easier to evaluate.

Trigonometric Identities and Derivatives

Knowing the derivatives of trigonometric functions like tan(θ) and sec(θ) is essential. For example, the derivative of tan(θ) is sec²(θ), which often appears in integrals and suggests substitution opportunities. Recognizing these relationships aids in solving integrals involving trigonometric expressions.

Evaluate the integrals in Exercises 41–60.55. ∫(from -π/4 to π/4)cosh(tanθ)sec²θ dθ