"Integral formula Carry out the following steps to derive the formula ∫ csch x dx = ln |tanh(x / 2)| + C (Theorem 7.6).


b. Use the identity for sinh(2u) to show that 2 / sinh(2u) = sech² u / tanh u."

Verified step by step guidance

1. Recall the double-angle identity for hyperbolic sine: \(\sinh(2u) = 2 \sinh u \cosh u\).
2. Start with the expression \(\frac{2}{\sinh(2u)}\) and substitute the identity: \(\frac{2}{2 \sinh u \cosh u} = \frac{1}{\sinh u \cosh u}\).
3. Express \(\frac{1}{\sinh u \cosh u}\) in terms of \(\tanh u\) and \(\operatorname{sech} u\) by rewriting the denominator: \(\sinh u = \frac{\tanh u}{\operatorname{sech} u}\) and \(\cosh u = \frac{1}{\operatorname{sech} u}\).
4. Use the definitions \(\tanh u = \frac{\sinh u}{\cosh u}\) and \(\operatorname{sech} u = \frac{1}{\cosh u}\) to rewrite the expression \(\frac{1}{\sinh u \cosh u}\) as \(\frac{\operatorname{sech}^2 u}{\tanh u}\).
5. Conclude that \(\frac{2}{\sinh(2u)} = \frac{\operatorname{sech}^2 u}{\tanh u}\), as required.

Key Concepts

Hyperbolic Functions and Their Identities

Hyperbolic functions like sinh, cosh, and tanh are analogs of trigonometric functions but for a hyperbola. Key identities, such as sinh(2u) = 2 sinh u cosh u, help simplify expressions and are essential for manipulating integrals involving hyperbolic functions.

Integration of Hyperbolic Functions

Integrating hyperbolic functions often involves substitution and using their identities to rewrite the integrand. For example, integrating csch x requires expressing it in terms of sinh x and applying logarithmic integration techniques to arrive at the formula ∫ csch x dx = ln |tanh(x/2)| + C.

Algebraic Manipulation of Hyperbolic Expressions

Proving identities like 2 / sinh(2u) = sech² u / tanh u requires careful algebraic manipulation using definitions: sech u = 1/cosh u and tanh u = sinh u / cosh u. Rearranging and substituting these expressions is crucial to verify the given identity.