Evaluate the integrals in Exercises 41–60.
41. ∫sinh(2x)dx

Verified step by step guidance

1. Recall the definition of the hyperbolic sine function: \(\sinh(x) = \frac{e^{x} - e^{-x}}{2}\).
2. Use the linearity of the integral to write \(\int \sinh(2x) \, dx\) as an integral involving exponentials: \(\int \sinh(2x) \, dx = \int \frac{e^{2x} - e^{-2x}}{2} \, dx\).
3. Split the integral into two separate integrals: \(\frac{1}{2} \int e^{2x} \, dx - \frac{1}{2} \int e^{-2x} \, dx\).
4. Integrate each exponential term using the formula \(\int e^{ax} \, dx = \frac{1}{a} e^{ax} + C\): so \(\int e^{2x} \, dx = \frac{1}{2} e^{2x} + C\) and \(\int e^{-2x} \, dx = -\frac{1}{2} e^{-2x} + C\).
5. Combine the results and simplify to express the integral back in terms of hyperbolic functions, then add the constant of integration \(C\).

Key Concepts

Hyperbolic Functions

Hyperbolic functions, such as sinh(x) and cosh(x), are analogs of trigonometric functions but based on hyperbolas. The function sinh(x) is defined as (e^x - e^(-x))/2 and has properties similar to sine, useful in integration and differentiation.

Integration of Hyperbolic Functions

Integrating hyperbolic functions involves using their definitions or known integral formulas. For example, the integral of sinh(ax) with respect to x is (1/a)cosh(ax) + C, where a is a constant, leveraging the derivative relationship between sinh and cosh.

Constant of Integration

When evaluating indefinite integrals, it is essential to include the constant of integration, denoted as C, because integration is the inverse of differentiation and the original function could differ by a constant.