Problem 49 - Thomas Calculus

Question

Evaluate the integrals in Exercises \(37-54\). $$ \int_{0}^{\pi / 2} \tan \frac{x}{2} d x $$

Step-by-Step Solution

Answer

The integral \( \int_{0}^{\pi/2} \tan \frac{x}{2} \, dx \) diverges.

Step 1: Recognize the Integral Form

The integral given is \( \int_{0}^{\pi / 2} \tan \frac{x}{2} \, dx \). This is an integral involving the tangent function with a half-angle argument.

Step 2: Use Trigonometric Identity

Note that \( \tan \frac{x}{2} = \frac{1 - \cos x}{\sin x} \). We can use this identity to simplify the integral.

Step 3: Rewrite the Integral

Substitute the identity into the integral to get \( \int_{0}^{\pi / 2} \frac{1 - \cos x}{\sin x} \, dx \), which can be separated into two integrals: \( \int_{0}^{\pi / 2} \frac{1}{\sin x} \, dx - \int_{0}^{\pi / 2} \frac{\cos x}{\sin x} \, dx \).

Step 4: Simplify Further

The first integral \( \int_{0}^{\pi / 2} \frac{1}{\sin x} \, dx \) is \( \int_{0}^{\pi / 2} \csc x \, dx \), and the second integral \( \int_{0}^{\pi / 2} \frac{\cos x}{\sin x} \, dx \) is \( \int_{0}^{\pi / 2} \cot x \, dx \).

Step 5: Solve the Integrals

The integral \( \int \csc x \, dx \) is \( \ln |\csc x - \cot x| + C \), and \( \int \cot x \, dx \) is \( \ln |\sin x| + C \). Evaluate both integrals from \(0\) to \(\pi/2\).

Step 6: Evaluate Definite Integrals

Substitute \( x = \pi/2 \) and \( x = 0 \) into \( \ln |\csc x - \cot x| \) and \( \ln |\sin x| \). For \( \int_{0}^{\pi/2} \csc x \, dx \), we have \( \ln |\csc (\pi/2)| - \cot (\pi/2)| = 0 \). For \( \int_{0}^{\pi/2} \cot x \, dx \), it is undefined at \(x = 0\), and the integral diverges.

Step 7: Conclusion

Since the integral \( \int_{0}^{\pi/2} \cot x \, dx \) diverges due to a singularity at the lower limit, the original integral diverges.

Key Concepts

Definite Integrals

In calculus, definite integrals are used to calculate the area under a curve from one point to another. This is represented as \( \int_{a}^{b} f(x) \ dx \), where \( a \) and \( b \) are the limits of integration and \( f(x) \) is the function under examination.
When dealing with trigonometric integrals, such as \( \int_{0}^{\pi / 2} \tan \frac{x}{2} \, dx \), the process involves finding the antiderivative of the trigonometric function.

Important steps include:

• Identifying the integral form and isolating any trigonometric identities that simplify the computation process.
• Determining whether the integral is convergent or divergent based on the behavior of the function over the specified interval.

Definite integrals can often yield interesting results depending on the function's behavior at the boundary points, which is critical in evaluating them correctly.

Trigonometric Identities

Trigonometric identities are equations involving trigonometric functions that are true for all values of the variable where both sides of the equality are defined. These identities are crucial in simplifying and solving integrals that involve trigonometric functions.

When integrating \( \int_{0}^{\pi / 2} \tan \frac{x}{2} \, dx \), the identity \( \tan \frac{x}{2} = \frac{1 - \cos x}{\sin x} \) is used to simplify the expression. By applying this identity, the integral is broken down into two simpler integrals:

• \( \int_{0}^{\pi / 2} \frac{1}{\sin x} \, dx \) or \( \int_{0}^{\pi / 2} \csc x \, dx \)
• \( \int_{0}^{\pi / 2} \frac{\cos x}{\sin x} \, dx \) or \( \int_{0}^{\pi / 2} \cot x \, dx \)

This transformation into other known integral forms makes the solution process more manageable and highlights the utility of trigonometric identities in integration.

Divergent Integrals

A divergent integral is one that does not converge to a finite value as it approaches its limits of integration. Recognizing divergent integrals is crucial because they indicate that the value of the integral extends to infinity or does not exist.
In the example \( \int_{0}^{\pi / 2} \tan \frac{x}{2} \, dx \), the integral \( \int_{0}^{\pi / 2} \cot x \, dx \) diverges due to a singularity at the lower limit, \( x = 0 \).

This singularity results in the value of the function \( \cot x \) tending towards infinity as \( x \) approaches zero.
This behavior highlights the importance of checking the convergence of each part of a separated integral,

• Identifying potential singularities within the limits of integration.
• Evaluating each integral carefully to understand the nature and outcome of divergence within the context of the problem.

Divergent integrals serve as a critical concept in calculus and require careful consideration during evaluation.