Question

In the following exercises, evaluate the definite integral. $$ \int_{\pi / 4}^{\pi / 3} \cot x d x $$

Short answer

\(\ln \left( \sqrt{\frac{3}{2}} \right)\)

Step-by-step solution

Recognize the Integral

The given integral is definite, and it requires evaluating \[\int_{\pi/4}^{\pi/3} \cot x \, dx.\]Cotangent is the reciprocal of tangent, defined as \(\cot x = \frac{\cos x}{\sin x}\). This suggests using a substitution to simplify.

Use Trig Identity for Integration

Observe that \(\cot x = \frac{\cos x}{\sin x}\) can be integrated using the natural logarithm property by employing substitution.Let \(u = \sin x\) so that \(du = \cos x \, dx\). Then the integral \(\int \cot x \, dx\) becomes \(\int \frac{du}{u}\). The solution of which is \(\ln |u| + C\).

Substitute Back to Original Variable

Substituting back \(u = \sin x\), the integral becomes \(\ln |\sin x| + C\).

Evaluate the Definite Integral Boundaries

Apply the boundaries \(\pi/4\) to \(\pi/3\) to evaluate:\[\left[\ln |\sin x|\right]_{\pi/4}^{\pi/3} = \ln |\sin(\pi/3)| - \ln |\sin(\pi/4)|\]\(\sin(\pi/3) = \frac{\sqrt{3}}{2}\) and \(\sin(\pi/4) = \frac{\sqrt{2}}{2}\).

Simplify the Expression

Plug in values into the expression:\[\ln \left( \frac{\sqrt{3}}{2} \right) - \ln \left( \frac{\sqrt{2}}{2} \right) = \ln \left( \frac{\sqrt{3}}{2} \times \frac{2}{\sqrt{2}} \right)\]Simplify using logarithm properties,\[= \ln \left( \frac{\sqrt{3}}{\sqrt{2}} \right) = \ln \left( \sqrt{\frac{3}{2}} \right)\]

Final Answer

Simplify further if necessary. The final evaluated definite integral result is:\[\ln \left( \sqrt{\frac{3}{2}} \right)\]

Key concepts

Trigonometric Integration

When dealing with integrals involving trigonometric functions, we refer to this process as trigonometric integration. Specifically, trigonometric integration frequently involves functions like sine, cosine, tangent, or their reciprocals, such as cotangent. In this particular exercise, we are presented with the integral of the cotangent function, \[\int \cot x \, dx\]This integral can be somewhat complex because it requires knowledge of the relationships between trigonometric functions.

• Cotangent is the reciprocal of tangent, which can be expressed as \(\cot x = \frac{\cos x}{\sin x}\).
• The reciprocal property is key to transforming the problem into a simpler form for integration.

Recognizing these identities allows us to use them effectively to simplify and solve integrals.

Natural Logarithm

The natural logarithm, denoted \( \ln \), plays a key role in integrating fractions where the variable of integration appears in the denominator. This arises when substituting in integrals involving cotangent. After transforming the integral using substitution, we arrive at:\[\int \frac{du}{u} = \ln |u| + C\]

• The natural logarithm helps us solve the integral involving a fraction \(\frac{1}{u}\).
• It emerges naturally through the process of integrating by substitution, when the derivative becomes \( \frac{1}{x} \).

Natural logarithms simplify expressing logarithmic results of integrals, making them fundamental when dealing with integrals that include fractions in the denominator.

Substitution Method

The substitution method is a powerful technique to simplify the integration process, particularly when dealing with complex trigonometric functions like cotangent. By substituting a part of the integral with a new variable, we can transform it into an easier problem.
In this exercise, by setting \( u = \sin x \), the integral of \( \cot x \) was transformed as follows:

• Set \( u = \sin x \) and differentiate to find \( du = \cos x \, dx \).
• The integral \( \int \cot x \, dx \) becomes \( \int \frac{du}{u} \), which simplifies the process significantly.

Substitution is a strategic tool that enables transforming challenging integrals into more manageable forms, ultimately solving the problem with greater ease.

Integration Bounds

When evaluating a definite integral, not only do we need to perform the integration, but we also need to consider the bounds of integration. The definite integral from \( \pi/4 \) to \( \pi/3 \) requires evaluating the solution at these limits:

• First, substitute the upper bound: \( \ln |\sin(\pi/3)| = \ln \left( \frac{\sqrt{3}}{2} \right) \).
• Then, substitute the lower bound: \( \ln |\sin(\pi/4)| = \ln \left( \frac{\sqrt{2}}{2} \right) \).
• Finally, subtract the lower bound result from the upper bound result to get the final answer.

Integration bounds define the scope within which the integral is calculated and ensure the result represents the exact area under the curve between these two points. This is the final step in establishing the specific value of the definite integral.