63–68. Definite integrals Evaluate the following definite integrals. Use Theorem 7.7 to express your answer in terms of logarithms.
∫₁ᵉ^² dx/x√(ln²x + 1)

Verified step by step guidance

1. Identify the integral to be evaluated: \(\displaystyle \int_1^{e^2} \frac{1}{x \sqrt{(\ln x)^2 + 1}} \, dx\).
2. Recognize that the integrand involves \(\ln x\) and its derivative. Consider the substitution \(t = \ln x\), which implies \(dt = \frac{1}{x} dx\).
3. Rewrite the integral in terms of \(t\): since \(x\) goes from 1 to \(e^2\), then \(t\) goes from \(\ln 1 = 0\) to \(\ln e^2 = 2\). The integral becomes \(\int_0^2 \frac{1}{\sqrt{t^2 + 1}} \, dt\).
4. Recall that \(\int \frac{1}{\sqrt{t^2 + 1}} \, dt\) is a standard integral whose antiderivative is \(\sinh^{-1}(t)\) or equivalently \(\ln(t + \sqrt{t^2 + 1})\) (this is Theorem 7.7, expressing inverse hyperbolic sine in terms of logarithms).
5. Evaluate the definite integral by substituting the limits \(t=0\) and \(t=2\) into the antiderivative \(\ln(t + \sqrt{t^2 + 1})\) and subtracting: \(\left. \ln(t + \sqrt{t^2 + 1}) \right|_0^2\).

Key Concepts

Definite Integrals

A definite integral calculates the net area under a curve between two specific limits. It is represented as ∫_a^b f(x) dx, where a and b are the lower and upper bounds. Evaluating definite integrals often involves finding an antiderivative and then applying the Fundamental Theorem of Calculus.

Substitution Method

The substitution method simplifies integrals by changing variables to transform the integral into a more manageable form. Typically, a substitution u = g(x) is chosen so that du replaces part of the integrand, making integration straightforward. This method is especially useful when the integrand contains composite functions.

Logarithmic Functions and Their Properties

Logarithmic functions, such as ln(x), are inverses of exponential functions and have properties that simplify expressions involving products, quotients, and powers. In integration, expressing answers in terms of logarithms often involves recognizing derivatives of ln(x) and using log rules to rewrite results clearly.