Question

The value of \(\int \frac{n\sqrt{\text{cosec}^2x^n-1}}{x^{(1-n)}}\,dx\) is ...

• A. \(log(sinx)+c\)
• B. \(log(sinx^n)+c\)
• C. \(log(cotx^n)+c\)
• D. \(log(cosx^n)+c\)

Hint:

Use the identity cosec squared minus one equals cot squared and substitute t = x^n.

Answer 1

The Correct Option is B

Step 1: Understanding the Concept:
Since \(\text{cosec}^2\theta - 1 = \cot^2\theta\), we have \(\sqrt{\text{cosec}^2x^n - 1} = \cot x^n\) for the usual range where \(\cot x^n > 0\).

Step 2: Key Formula or Approach:
\(\frac{n}{x^{1-n}} = n x^{n-1}\), and \(\frac{d}{dx}(x^n) = nx^{n-1}\).

Step 3: Detailed Explanation:
The integral is \(\int nx^{n-1}\cot(x^n)\,dx\).
Put \(t = x^n\), so \(dt = nx^{n-1}dx\).
\[ \int \cot t\,dt = \log|\sin t| + c = \log(\sin x^n) + c \]
The answer is \(\log(\sin x^n)\), not \(\log(\sin x)\), because the substitution keeps the power inside the sine.

Final Answer:
The value is \(\log(\sin x^n) + c\), option (B). \[ \boxed{\log(\sin x^n)+c} \]