37–56. Integrals Evaluate each integral.
∫ (cosh z) / (sinh² z) dz

Verified step by step guidance

1. Recognize that the integral involves hyperbolic functions: \(\cosh z\) and \(\sinh z\). Recall the derivatives: \(\frac{d}{dz} \sinh z = \cosh z\) and \(\frac{d}{dz} \cosh z = \sinh z\).
2. Rewrite the integral as \(\int \frac{\cosh z}{\sinh^{2} z} \, dz = \int \cosh z \cdot \sinh^{-2} z \, dz\) to see the structure more clearly.
3. Use substitution by letting \(u = \sinh z\). Then, \(du = \cosh z \, dz\), which means \(\cosh z \, dz = du\).
4. Substitute into the integral to get \(\int u^{-2} \, du\), which simplifies the integral to a power function of \(u\).
5. Integrate \(\int u^{-2} \, du\) using the power rule for integrals, then substitute back \(u = \sinh z\) to express the answer in terms of \(z\).

Key Concepts

Hyperbolic Functions

Hyperbolic functions, such as sinh(z) and cosh(z), are analogs of trigonometric functions but based on exponential functions. They satisfy identities like cosh²(z) - sinh²(z) = 1, which are useful in simplifying expressions and integrals involving these functions.

Integration Techniques for Rational Functions

Integrals involving ratios of functions often require substitution or rewriting the integrand to a simpler form. Recognizing derivatives within the integrand, such as identifying if the numerator is the derivative of the denominator, helps in applying substitution effectively.

Substitution Method in Integration

The substitution method involves changing variables to simplify an integral. By letting u equal a function inside the integral (e.g., u = sinh(z)), the integral can be transformed into a more straightforward form, making it easier to evaluate.