Problem 35 - Thomas Calculus

Question

Evaluate each integral in Exercises \(1-36\) by using a substitution to reduce it to standard form. $$ \int_{1}^{e^{\pi / 3}} \frac{d x}{x \cos (\ln x)} $$

Step-by-Step Solution

Answer

The value of the integral is \(\ln(2 + \sqrt{3})\).

Step 1: Identify the Substitution

Given the integral \(\int_{1}^{e^{\pi / 3}} \frac{d x}{x \cos (\ln x)}\), we can use substitution to solve it. Let \(u = \ln(x)\). Then, the differential \(du = \frac{1}{x}dx\).

Step 2: Change Limits of Integration

Substitute the limits of integration. When \(x = 1\), \(u = \ln(1) = 0\). When \(x = e^{\pi/3}\), \(u = \ln(e^{\pi/3}) = \pi/3\). Thus, the new limits for \(u\) are from \(0\) to \(\pi/3\).

Step 3: Rewrite the Integral

With the substitution \(u = \ln(x)\), the integral becomes \(\int_{0}^{\pi/3} \frac{du}{\cos(u)}\). Since \(\frac{1}{\cos(u)} = \sec(u)\), this transforms to \(\int_{0}^{\pi/3} \sec(u) \, du\).

Step 4: Integrate the New Form

The integral \(\int \sec(u) \, du = \ln|\sec(u) + \tan(u)| + C\). Evaluate this from \(u = 0\) to \(u = \pi/3\).

Step 5: Evaluate the Definite Integral

Find the value of \(\ln|\sec(u) + \tan(u)|\) at the limits. At \(u = \pi/3\), \(\sec(\pi/3) = 2\) and \(\tan(\pi/3) = \sqrt{3}\), so \(\ln|2 + \sqrt{3}|\). At \(u = 0\), \(\sec(0) = 1\) and \(\tan(0) = 0\), so \(\ln|1| = 0\). Subtract to find: \(\ln|2 + \sqrt{3}| - 0 = \ln(2 + \sqrt{3})\).

Key Concepts

Understanding Definite Integrals

A definite integral is a fundamental concept in integral calculus that represents the signed area under a curve between two points on the x-axis. When you evaluate a definite integral, you determine the accumulated value of a function between these two limits. In terms of notation, a definite integral is usually written as \[ \int_{a}^{b} f(x) \, dx \] where \(a\) and \(b\) are the lower and upper limits of integration, respectively.

• It gives the total change of the quantity represented by the function \(f(x)\) over the interval \([a, b]\).
• The result of a definite integral is a number, unlike indefinite integrals which yield functions.

To solve a definite integral, you first find an antiderivative of the function, and then compute the difference between the values of this antiderivative at the upper and lower bounds. In practice, definite integrals are often used to calculate areas, volumes, or any quantity that requires accumulation over an interval.

The Substitution Method in Calculus

The substitution method is a technique used to simplify complex integrals, making them easier to evaluate. It's akin to the reverse process of the chain rule in differentiation. Through substitution, complicated integrals can often be transformed into simpler standard forms that are easier to solve.

Here's how the substitution method works:

• Choose a substitution: Identify a part of the integrand (the function you are integrating) that can be replaced with a single variable, typically \(u\). This choice often involves a function and its derivative.
• Compute the differential: Once you've selected \(u\), find \(\frac{du}{dx}\) to substitute \(dx\) in terms of \(du\).
• Adjust the limits: If you're dealing with a definite integral, change the limits of integration to match the new variable.
• Rewrite and integrate: Replace all instances of the original variable and its differential with the substitution variable and its differential, then integrate the simpler expression.

Using substitution simplifies the problem: our original problem turned into \[ \int_{0}^{\pi/3} \sec(u) \, du \] a problem easily tackled with known antiderivatives.

Basics of Trigonometric Integration

Trigonometric integration involves integrating functions that include trigonometric functions like sine, cosine, tangent, and secant. These functions can often make integrals more challenging due to their periodic nature and complex derivatives. But with a little understanding, they become more approachable.

• Common Trigonometric Integrals: Some trigonometric integrals, like \(\int \sin(x) \, dx = -\cos(x) + C\) or \(\int \cos(x) \, dx = \sin(x) + C\), have straightforward antiderivatives.
• Advanced Forms: Integrals involving secant, such as \(\int \sec(x) \, dx\), require more sophisticated methods or standard results, yielding \(\ln|\sec(x) + \tan(x)| + C\).
• Subtraction and Addition Rules: Many trigonometric integrals can be solved by basic identities or algebraic manipulation, simplifying them into forms recognizable by their antiderivatives.

The crucial step lies in recognizing the right trigonometric identity or transformation for simplification. In our integral, the use of substitution changed a logarithmic function into a simple trigonometric one \(\sec(u)\), which is known and manageable.