37–56. Integrals Evaluate each integral.
∫ sinh²z dz (Hint: Use an identity.)

Verified step by step guidance

1. Recall the hyperbolic identity for sinh squared: \(\sinh^{2}z = \frac{\cosh(2z) - 1}{2}\). This will help simplify the integral.
2. Rewrite the integral using the identity: \(\int \sinh^{2}z \, dz = \int \frac{\cosh(2z) - 1}{2} \, dz\).
3. Split the integral into two separate integrals: \(\int \frac{\cosh(2z)}{2} \, dz - \int \frac{1}{2} \, dz\).
4. Integrate each term separately. For the first term, use the substitution \(u = 2z\) so that \(du = 2 \, dz\), and for the second term, integrate the constant.
5. After integrating, substitute back if needed and combine the results. Don't forget to add the constant of integration \(C\) at the end.

Key Concepts

Hyperbolic Functions

Hyperbolic functions, such as sinh(z) and cosh(z), are analogs of trigonometric functions but based on hyperbolas. They have unique properties and identities useful for integration, like sinh²(z) which can be expressed in terms of cosh(2z). Understanding their definitions and relationships is essential for solving integrals involving these functions.

Hyperbolic Identities

Hyperbolic identities simplify expressions involving hyperbolic functions. For example, the identity sinh²(z) = (cosh(2z) - 1)/2 helps transform the integral into a more manageable form. Recognizing and applying these identities allows for easier integration by converting powers of hyperbolic functions into sums or differences.

Integration Techniques

Integration techniques such as substitution and using standard integral formulas are crucial. After applying the hyperbolic identity, the integral often reduces to integrating cosh(2z) and constants, which have straightforward antiderivatives. Mastery of these techniques enables efficient evaluation of integrals involving hyperbolic functions.