Question

Evaluate: \( \int \text{cosec}\,x(\text{cosec}\,x + \cot x) \,dx \).

Hint:

Always simplify the integrand before attempting to integrate. Recognizing standard integral forms like \( \int \text{cosec}^2 x \,dx \) and \( \int \text{cosec}\,x \cot x \,dx \) is crucial for quickly solving such problems. Memorizing these fundamental formulas saves a lot of time in exams.

Answer 1

Step 1: Understanding the Concept:
This problem involves integrating a trigonometric function. The strategy is to simplify the integrand and then apply standard integration formulas.
Step 2: Key Formula or Approach:
The key is to use the distributive property to expand the integrand and then use the following standard integrals:
1. \( \int \text{cosec}^2 x \,dx = -\cot x + C \)
2. \( \int \text{cosec}\,x \cot x \,dx = -\text{cosec}\,x + C \)
Step 3: Detailed Explanation:
First, expand the expression inside the integral:
\[ \int \text{cosec}\,x(\text{cosec}\,x + \cot x) \,dx = \int (\text{cosec}^2 x + \text{cosec}\,x \cot x) \,dx \] Now, we can split the integral into two parts using the sum rule for integration:
\[ \int \text{cosec}^2 x \,dx + \int \text{cosec}\,x \cot x \,dx \] Apply the standard integration formulas to each part:
For the first part: \( \int \text{cosec}^2 x \,dx = -\cot x \).
For the second part: \( \int \text{cosec}\,x \cot x \,dx = -\text{cosec}\,x \).
Combining the results and adding the constant of integration, \( C \):
\[ -\cot x - \text{cosec}\,x + C \] Step 4: Final Answer:
The evaluation of the integral is \( -\cot x - \text{cosec}\,x + C \).