Problem 15 - Calculus 2 Course in Mathematics for the IIT-JEE and Other Engineering Exams

Question

\(\int \operatorname{coscc} x d x=\log (\operatorname{cosec} x-\cot x)=\log \tan \frac{x}{2}\)

Step-by-Step Solution

Answer

The integral of \( \operatorname{cosec} x \) is \( \log (\operatorname{cosec} x - \cot x) + C = \log \tan \frac{x}{2} + C \).

Step 1: Understand the problem

We need to evaluate the integral of \( \operatorname{cosec} x \), which is the cosecant function, and show that it equals \( \log (\operatorname{cosec} x - \cot x) = \log \tan \frac{x}{2} \). This will require integrating \( \operatorname{cosec} x \) and then verifying the equivalence with the given logarithmic expressions.

Step 2: Integrate the cosecant function

To integrate \( \operatorname{cosec} x \), we use the identity \( \operatorname{cosec} x = \frac{1}{\sin x} \). Typically, the integral of cosecant is given as:\[\int \operatorname{cosec} x \ dx = \log |\operatorname{cosec} x - \cot x| + C\]where \( C \) is the constant of integration.

Step 3: Verifying the equivalence with logarithm form

Now we need to verify that \( \log \left( \operatorname{cosec} x - \cot x \right) = \log \tan \frac{x}{2} \). This is a known trigonometric identity involving half-angle formulas and can be shown by using half-angle identities. We can use the identity \( \tan \frac{x}{2} = \sqrt{\frac{1 - \cos x}{1 + \cos x}} \) and trigonometric simplifications to verify the equivalence. For simplification,\[ \operatorname{cosec} x - \cot x = \frac{1}{\sin x} - \frac{\cos x}{\sin x} = \frac{1 - \cos x}{\sin x}\] then link it with half-angle identity via\[\tan \frac{x}{2} = \frac{\sin x}{1 + \cos x} \equiv \frac{1 - \cos x}{\sin x} \]

Step 4: Conclusion and final check

Through calculations of integral and identity checks using known trigonometric identities, we conclude that:\[\int \operatorname{cosec} x \ dx = \log (\operatorname{cosec} x - \cot x) = \log \tan \frac{x}{2} + C\]Thus verifying both of the originally given expressions as equivalent.

Key Concepts

Cosecant Integral

The trigonometric function cosecant, denoted as \( \operatorname{cosec} x \), is the reciprocal of the sine function. When integrating this function, you approach it as \( \operatorname{cosec} x = \frac{1}{\sin x} \). This integral is known for its involvement in the expression \( \log |\operatorname{cosec} x - \operatorname{cot} x| + C \), where \( C \) is an integration constant.

To properly tackle this problem, remember that the integral of \( \operatorname{cosec} x \) is not straightforward and generally requires the application of advanced techniques. These include using substitution to simplify the trigonometric identities involved.

If you substitute and simplify correctly, you will transform the integral into its logarithmic form involving cosecant and cotangent, leading to the result \( \log |\operatorname{cosec} x - \operatorname{cot} x| \). This simplification is crucial for the integral to precisely match known results in calculus.

Trigonometric Identities

Trigonometric identities like the ones used in \( \log (\operatorname{cosec} x - \operatorname{cot} x) \) simplify expressions involving angles and sides of triangles. These identities are essential tools for verifying equivalences and transforming expressions.

In this exercise, the identity that relates \( \tan \frac{x}{2} \) to the combination \( \operatorname{cosec} x - \operatorname{cot} x \) was leveraged. The commonly used identities include:

• \( \operatorname{sin}^2 \theta + \operatorname{cos}^2 \theta = 1 \)
• \( 1 + \operatorname{tan}^2 \theta = \operatorname{sec}^2 \theta \)
• \( \operatorname{tan} \left(\frac{x}{2}\right) = \sqrt{\frac{1 - \operatorname{cos} x}{1 + \operatorname{cos} x}} \)

These identities allow conversions that simplify complex expressions and solve equations that seem daunting at first. By substituting these identities appropriately, the initial complex expressions are reduced, verifying equality between various trigonometric forms.

Half-Angle Formulas

Half-angle formulas are a powerful set of trigonometric identities used to express functions of half angles. They are valuable for dividing an angle into two equal parts, which simplifies the integration of trigonometric functions.

A crucial half-angle identity used in the context of this exercise is \( \tan \frac{x}{2} = \frac{1 - \cos x}{\sin x} \). This helps to bridge the gap between different trigonometric expressions, enabling a deeper understanding of the equivalencies involved.

This particular identity connects with \( \tan \frac{x}{2} = \frac{\operatorname{sin} x}{1 + \operatorname{cos} x} \), as shown in the integration process. Using half-angle formulas not only aids in simplification but also assists in revealing hidden relationships between trigonometric functions.

By understanding these formulas, you gain tools that can be applied to various integral evaluations and transformations, making trigonometric integration more manageable and illuminating the path to the solution.