Evaluate the integrals in Exercises 41–60.
59. ∫(from -ln2 to 0)cosh²(x/2) dx

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Recall the definition of the hyperbolic cosine function: \(\cosh(x) = \frac{e^x + e^{-x}}{2}\). However, for integration, it is often easier to use the identity involving \(\cosh^2\).

Use the identity \(\cosh^2(u) = \frac{1 + \cosh(2u)}{2}\) to rewrite the integrand. Here, let \(u = \frac{x}{2}\), so the integral becomes \(\int_{-\ln 2}^0 \cosh^2\left(\frac{x}{2}\right) dx = \int_{-\ln 2}^0 \frac{1 + \cosh(x)}{2} dx\).

Split the integral into two simpler integrals: \(\int_{-\ln 2}^0 \frac{1}{2} dx + \int_{-\ln 2}^0 \frac{\cosh(x)}{2} dx\).

Integrate each part separately: the integral of \(\frac{1}{2}\) with respect to \(x\) is \(\frac{x}{2}\), and the integral of \(\frac{\cosh(x)}{2}\) with respect to \(x\) is \(\frac{\sinh(x)}{2}\).

Evaluate the resulting expression \(\left[ \frac{x}{2} + \frac{\sinh(x)}{2} \right]\) at the limits \(x = -\ln 2\) and \(x = 0\), and subtract to find the value of the definite integral.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Hyperbolic Cosine Function (cosh)

The hyperbolic cosine function, cosh(x), is defined as (e^x + e^(-x))/2. It shares properties similar to the cosine function but relates to hyperbolic geometry. Understanding its definition and behavior is essential for integrating expressions involving cosh.

Integration of Hyperbolic Functions

Integrating hyperbolic functions often involves using identities or rewriting the function in exponential form. For example, cosh²(x) can be expressed using the identity cosh²(x) = (1 + cosh(2x))/2, which simplifies the integral into more manageable terms.

Definite Integrals and Limits of Integration

A definite integral calculates the net area under a curve between two specific points. Evaluating from -ln(2) to 0 requires substituting these limits into the antiderivative after integration, ensuring careful handling of logarithmic and exponential expressions.

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