37–56. Integrals Evaluate each integral.
∫ tanh²x dx (Hint: Use an identity.)

Verified step by step guidance

1. Recall the identity for hyperbolic tangent squared: \(\tanh^{2}x = 1 - \sech^{2}x\). This will help simplify the integral.
2. Rewrite the integral using the identity: \(\int \tanh^{2}x \, dx = \int (1 - \sech^{2}x) \, dx\).
3. Split the integral into two separate integrals: \(\int 1 \, dx - \int \sech^{2}x \, dx\).
4. Integrate each term separately: The integral of 1 with respect to \(x\) is \(x\), and the integral of \(\sech^{2}x\) is \(\tanh x\).
5. Combine the results to write the integral as \(x - \tanh x + C\), where \(C\) is the constant of integration.

Key Concepts

Hyperbolic Functions and Their Identities

Hyperbolic functions like tanh(x) are analogs of trigonometric functions but based on exponential functions. Key identities, such as tanh²x + sech²x = 1, help simplify expressions and integrals involving these functions.

Integration Techniques Using Identities

Using algebraic or trigonometric/hyperbolic identities can transform complicated integrals into simpler forms. For example, rewriting tanh²x using an identity allows the integral to be expressed in terms of easier-to-integrate functions.

Basic Integration of Hyperbolic Functions

Integrals involving hyperbolic functions often reduce to standard forms, such as ∫sech²x dx = tanh x + C. Recognizing these standard integrals is essential for solving problems efficiently.