37–56. Integrals Evaluate each integral.
∫ cosh 2x dx

Verified step by step guidance

1. Recall the definition of the hyperbolic cosine function: \(\cosh x = \frac{e^x + e^{-x}}{2}\).
2. Use the double angle identity for hyperbolic cosine: \(\cosh 2x = 2\cosh^2 x - 1\) or express \(\cosh 2x\) in terms of exponentials as \(\frac{e^{2x} + e^{-2x}}{2}\).
3. Rewrite the integral using the exponential form: \(\int \cosh 2x \, dx = \int \frac{e^{2x} + e^{-2x}}{2} \, dx\).
4. Split the integral into two simpler integrals: \(\frac{1}{2} \int e^{2x} \, dx + \frac{1}{2} \int e^{-2x} \, dx\).
5. Integrate each term using the formula \(\int e^{ax} \, dx = \frac{1}{a} e^{ax} + C\), then combine the results and add the constant of integration.

Key Concepts

Hyperbolic Functions

Hyperbolic functions, such as cosh(x) and sinh(x), are analogs of trigonometric functions but based on exponential functions. Specifically, cosh(x) = (e^x + e^(-x))/2. Understanding their definitions and properties is essential for integrating expressions involving cosh.

Integration of Exponential Functions

Since hyperbolic functions are expressed in terms of exponentials, integrating cosh(2x) involves integrating exponential functions like e^(2x). The integral of e^(ax) with respect to x is (1/a)e^(ax) + C, which helps in solving the integral of cosh(2x).

Use of Substitution in Integration

When integrating functions like cosh(2x), substitution is useful to simplify the integral. Letting u = 2x transforms the integral into a simpler form, allowing the use of standard integration rules and then substituting back to the original variable.