Problem 3 - Calculus 2 Course in Mathematics for the IIT-JEE and Other Engineering Exams

Question

Integrals of the form: \(\int \cot ^{m} x \operatorname{cosec}^{n} x d x\)Rule (i) If \(m\) is even or odd integer and \(n\) is even positive integer then put \(\cot x=t\). (ii) If \(m\) is odd positive integer and \(n \notin\) even positive integer then put cosec \(x=t\).(iii) I \(m=0\) and \(n=2 r+1 \forall r \in N\), then \(\int \operatorname{cosec}^{2 r+1} x d x=\int \operatorname{cosec}^{2 r-1} x \operatorname{cosec}^{2} x d x\), then integrate by parts taking \(\operatorname{cosec}^{2} x\) as second function.

Step-by-Step Solution

Answer

Identify \( m \) and \( n \), choose substitution, integrate, back-substitute.

Step 1: Identify the form of the integral

We are given an integral in the form \( \int \cot^m x \operatorname{cosec}^n x \, dx \). We need to determine the values of \( m \) and \( n \) and apply the corresponding rule for simplification.

Step 2: Choose the appropriate substitution

According to the rules:- If \( m \) is even or odd integer and \( n \) is even positive integer, substitute \( \cot x = t \).- If \( m \) is an odd positive integer and \( n otin\) even positive integer, substitute \( \operatorname{cosec} x = t \).- If \( m = 0 \) and \( n = 2r + 1 \) for \( r \in \mathbb{N} \), use integration by parts with \( \operatorname{cosec}^2 x \) as the second function.Determine \( m \) and \( n \) in your integral and choose the substitution accordingly.

Step 3: Apply the substitution and integrate

Proceed with the substitution chosen in Step 2. For instance:- If \( \cot x = t \), then \( -\operatorname{cosec}^2 x \, dx = dt \).- If \( \operatorname{cosec} x = t \), then \( -\cot x \operatorname{cosec} x \, dx = dt \).Substitute and integrate the resulting expression.

Step 4: Back-substitute and simplify

After integrating, replace the variable \( t \) with the trigonometric function originally used for substitution, to express the solution in terms of \( x \).

Key Concepts

Substitution Method

The substitution method is an essential technique in calculus to simplify complex integrals, making them more manageable. When integrating trigonometric expressions such as \( \int \cot^m x \operatorname{cosec}^n x \, dx \), the key is to identify a function within the integral to replace with a single variable, typically \( t \). This transformation simplifies the integral, facilitating easier computation.To use this method:

• Recognize patterns or forms in the integral that match common substitution types.
• Apply the appropriate substitution. For example, if \( \cot x = t \), then calculate \( dt = -\operatorname{cosec}^2 x \, dx \).
• Transform the integral into a simpler form in terms of \( t \), allowing you to integrate using basic techniques.
• Always back-substitute your original variable to express the solution in terms of \( x \) once integration is complete.

Utilizing the substitution method effectively reduces complexity and converts challenging trigonometric integrals into more straightforward problems.

Integration by Parts

Integration by parts is another powerful tool in the integration toolbox, especially for products of functions. It is particularly useful for integrals like \( \int \operatorname{cosec}^{2r+1} x \, dx \), where straightforward substitution is not practical.The formula for integration by parts is:\[\int u \, dv = uv - \int v \, du\]Here's how to apply it:

• Identify two parts of your integral, \( u \) and \( dv \). A strategic choice is often necessary, depending on the functions involved.
• Differentiate \( u \) to find \( du \), and integrate \( dv \) to get \( v \).
• Substitute these expressions back into the integration by parts formula, simplifying the remaining terms if possible.
• The choice of \( u \) is crucial: it should be something that simplifies when differentiated (e.g., \( \operatorname{cosec}^2 x \) in this exercise).

Integration by parts often splits a complex problem into easier parts, making it an invaluable technique in tackling integrals involving products.

Integration Techniques

Integration is all about the art of simplifying, and mastering integration techniques can significantly aid in solving even the most daunting integrals. Some key approaches include substitution and integration by parts, as previously discussed, but there's more to explore.Other integration techniques to consider include:

• Partial fraction decomposition: Useful for breaking down rational functions into simpler fractions, which are easier to integrate individually.
• Trigonometric identities: Simplifying integrals by using identities like \( \sin^2 x + \cos^2 x = 1 \) or converting products into sums.
• Trigonometric substitution: Applying trigonometric identities to turn algebraic expressions into trigonometric integrals.

Knowing when and how to apply these techniques requires practice and intuition. However, once mastered, these skills empower you to tackle a wide range of integral-related challenges with confidence and precision.