Evaluate the integrals in Exercises 39–56.
56. ∫sec(x)dx/√(ln(sec(x)+tan(x)))

Verified step by step guidance

1. Start by examining the integral: \(\int \frac{\sec(x)}{\sqrt{\ln(\sec(x) + \tan(x))}} \, dx\). Notice the expression inside the logarithm, \(\sec(x) + \tan(x)\), which suggests a substitution involving this term.
2. Let’s set \(u = \ln(\sec(x) + \tan(x))\). This substitution is motivated because the denominator contains \(\sqrt{u}\), and the derivative of \(u\) will likely simplify the integral.
3. Differentiate \(u\) with respect to \(x\): \(\frac{du}{dx} = \frac{1}{\sec(x) + \tan(x)} \cdot \frac{d}{dx}(\sec(x) + \tan(x))\). Recall that \(\frac{d}{dx} \sec(x) = \sec(x) \tan(x)\) and \(\frac{d}{dx} \tan(x) = \sec^2(x)\).
4. Calculate \(\frac{d}{dx}(\sec(x) + \tan(x)) = \sec(x) \tan(x) + \sec^2(x) = \sec(x)(\tan(x) + \sec(x))\). Substitute back into \(\frac{du}{dx}\): \(\frac{du}{dx} = \frac{\sec(x)(\tan(x) + \sec(x))}{\sec(x) + \tan(x)}\).
5. Notice that \(\tan(x) + \sec(x)\) and \(\sec(x) + \tan(x)\) are the same, so they cancel out, leaving: \(\frac{du}{dx} = \sec(x)\). Therefore, \(du = \sec(x) \, dx\), which matches the numerator of the integral. This means the integral can be rewritten in terms of \(u\) as: \(\int \frac{1}{\sqrt{u}} \, du\). From here, you can proceed to integrate with respect to \(u\).

Key Concepts

Integration Techniques involving Substitution

Substitution is a method used to simplify integrals by changing variables, often turning a complicated integral into a more manageable form. Recognizing an inner function and its derivative within the integral is key to applying substitution effectively.

Derivatives and Identities of Secant and Tangent Functions

Understanding the derivatives and relationships between sec(x) and tan(x) is crucial, as their derivatives are interconnected (d/dx sec(x) = sec(x)tan(x), d/dx tan(x) = sec^2(x)). These identities help in simplifying expressions and recognizing substitution candidates.

Logarithmic Functions and their Properties

Logarithmic functions, such as ln(sec(x) + tan(x)), often appear inside integrals. Knowing how to differentiate and manipulate logarithms, including the chain rule and properties of logarithms, aids in simplifying the integral and identifying substitution strategies.