91Ó°ÊÓ

Hyperbolic functions are like trigonometric functions, but for hyperbolas instead of circles. They are used frequently in calculus, especially in integrals and differential equations. These functions can be defined using exponential functions. For instance, the hyperbolic sine function, \( \sinh(x) = \frac{e^x - e^{-x}}{2} \), and the hyperbolic cosine function, \( \cosh(x) = \frac{e^x + e^{-x}}{2} \). Similar to trigonometric identities, hyperbolic functions have identities like \( \cosh^2(x) - \sinh^2(x) = 1 \).
The hyperbolic tangent function, \( \tanh(x) = \frac{\sinh(x)}{\cosh(x)} \), is the function we focus on in this exercise. It behaves similarly to the tangent function in trigonometry and reflects unique properties due to its exponential nature, making it very smooth and continuous everywhere.

Integration by substitution is a method used in calculus to simplify integrations by transforming complex functions into simpler ones. Similar to the chain rule for differentiation, substitution can often make seemingly difficult integrals more manageable.
In this problem, we encounter \( \tanh(2x) \), which can be made simpler by letting \( u = 2x \). Hence, the differentials change because \( du = 2dx \) or \( dx = \frac{1}{2}du \). This technique often requires changing the limits of integration as well since we express everything in terms of \( u \).
This step is crucial as it transforms the integral into a more recognizable form that is easier to integrate, such as when integrating hyperbolic functions.

Solving calculus problems involves a structured approach, often starting from analyzing the form of the function we are dealing with. Here, the integral \( \int \tanh(2x) dx \) requires understanding hyperbolic functions and the use of substitution for simplification. Calculus problems can sometimes seem daunting, but breaking them down into smaller steps, such as applying known identities and substitution, can make them solvable.
Many calculus problems require checking if a simpler function is hiding within a more complex integrand. In this case, realizing that \( \tanh(2x) \) relates to \( \tanh(u) \) shows that converting variables and changing limits can reveal straightforward solutions through known results of hyperbolic functions.

When integrating, especially using substitution, it is essential to adjust the limits of integration accordingly. Here, since we substituted \( u = 2x \), we needed to change the bounds as well. The lower limit changes from \( x = 0 \) to \( u = 0 \), and the upper limit changes from \( x = \ln 2 \) to \( u = 2 \ln 2 \).
Properly transforming the limits ensures the integral retains the same original range and dependencies. Ignoring these changes can lead to an incorrect evaluation of the integral. After substitution and adjusting the integrand and limits, one can carry on with integrating the new, simpler expression.