Evaluating the Definite Integral of ln(tan(x/2))

We are asked to evaluate the definite integral: \( \int^\pi _0 \ln\left(\tan\frac{x}{2}\right) dx \). Let this integral be denoted by \(I\).

This integral can be solved effectively using a property of definite integrals, often referred to as the King's Property or Property 4:

For a function \(f(x)\), the definite integral over the interval \([0, a]\) satisfies:

\( \int_0^a f(x) dx = \int_0^a f(a-x) dx \)

In our case, the upper limit is \(a = \pi\), and the integrand is \(f(x) = \ln\left(\tan\frac{x}{2}\right)\). Let's apply the property by substituting \(x\) with \(a-x = \pi - x\):

\( I = \int^\pi _0 \ln\left(\tan\frac{x}{2}\right) dx \)

Using the property, we get:

\( I = \int^\pi _0 \ln\left(\tan\frac{\pi - x}{2}\right) dx \)

Simplify the argument of the tangent function:

\( \frac{\pi - x}{2} = \frac{\pi}{2} - \frac{x}{2} \)

So, the integral becomes:

\( I = \int^\pi _0 \ln\left(\tan\left(\frac{\pi}{2} - \frac{x}{2}\right)\right) dx \)

Recall the trigonometric identity: \( \tan\left(\frac{\pi}{2} - \theta\right) = \cot\theta \). Applying this identity with \(\theta = \frac{x}{2}\):

\( \tan\left(\frac{\pi}{2} - \frac{x}{2}\right) = \cot\frac{x}{2} \)

Substituting this back into the integral expression for \(I\):

\( I = \int^\pi _0 \ln\left(\cot\frac{x}{2}\right) dx \)

Now we have two expressions for the definite integral \(I\):

1. \( I = \int^\pi _0 \ln\left(\tan\frac{x}{2}\right) dx \) (Original integral)
2. \( I = \int^\pi _0 \ln\left(\cot\frac{x}{2}\right) dx \) (After applying the property)

Let's add these two expressions together:

\( I + I = \int^\pi _0 \ln\left(\tan\frac{x}{2}\right) dx + \int^\pi _0 \ln\left(\cot\frac{x}{2}\right) dx \)

\( 2I = \int^\pi _0 \left[\ln\left(\tan\frac{x}{2}\right) + \ln\left(\cot\frac{x}{2}\right)\right] dx \)

Using the property of logarithms: \( \ln a + \ln b = \ln(ab) \):

\( 2I = \int^\pi _0 \ln\left(\tan\frac{x}{2} \cdot \cot\frac{x}{2}\right) dx \)

Recall the trigonometric identity: \( \tan\theta \cdot \cot\theta = 1 \). Applying this identity with \(\theta = \frac{x}{2}\):

\( \tan\frac{x}{2} \cdot \cot\frac{x}{2} = 1 \)

Substituting this back into the integral:

\( 2I = \int^\pi _0 \ln(1) dx \)

The natural logarithm of 1 is 0: \( \ln(1) = 0 \).

\( 2I = \int^\pi _0 0 \, dx \)

The definite integral of 0 over any interval is 0:

\( 2I = 0 \)

Solving for \(I\):

\( I = \frac{0}{2} = 0 \)

Thus, the value of the definite integral \( \int^\pi _0 \ln\left(\tan\frac{x}{2}\right) dx \) is 0.

Revision Table: Definite Integral Evaluation

Review the key steps involved in solving this definite integral problem.

• Identify the definite integral: \( \int^\pi _0 \ln\left(\tan\frac{x}{2}\right) dx \).
• Recognize the applicability of the property \( \int_0^a f(x) dx = \int_0^a f(a-x) dx \).
• Apply the property with \(a = \pi\) to get a new form of the integral.
• Use trigonometric identities like \( \tan(\frac{\pi}{2} - \theta) = \cot\theta \).
• Add the original and the transformed integrals.
• Use logarithmic properties like \( \ln a + \ln b = \ln(ab) \).
• Use trigonometric identities like \( \tan\theta \cdot \cot\theta = 1 \).
• Evaluate the resulting simple integral.

Additional Information: Properties of Definite Integrals and Logarithms

Understanding properties of definite integrals and logarithms is crucial for solving such problems. The King's Property (Property 4) is particularly useful when the integrand involves functions that change their form predictably under the transformation \(x \to a-x\), often leading to simplification when combined with the original integral.

Key Properties Used:

• Definite Integral Property: \( \int_0^a f(x) dx = \int_0^a f(a-x) dx \)
• Logarithm Property: \( \ln a + \ln b = \ln(ab) \)
• Logarithm Property: \( \ln(1) = 0 \)
• Trigonometric Identity: \( \tan\left(\frac{\pi}{2} - \theta\right) = \cot\theta \)
• Trigonometric Identity: \( \tan\theta \cdot \cot\theta = 1 \)

These properties work together to transform a seemingly complex integral into a trivial one.