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

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Recall the identity for hyperbolic tangent squared: \(\tanh^{2}x = 1 - \sech^{2}x\). This will help simplify the integral.

Rewrite the integral using the identity: \(\int \tanh^{2}x \, dx = \int (1 - \sech^{2}x) \, dx\).

Split the integral into two separate integrals: \(\int 1 \, dx - \int \sech^{2}x \, dx\).

Integrate each term separately: The integral of 1 with respect to \(x\) is \(x\), and the integral of \(\sech^{2}x\) is \(\tanh x\).

Combine the results to write the integral as \(x - \tanh x + C\), where \(C\) is the constant of integration.

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주요 개념

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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.

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