Question 12 of 33intermediate🔧 ApplyShort Answer2 marks

Integrate the functions in Exercises 1 to 37: cot⁡\cot x log⁡sin⁡\log \sin x

Correct Answer

To find the integral of cot x log sin x, we use the substitution method.

Let t = log sin x. Differentiating both sides with respect to x: dt = (1/sin x) · cos x dx = cot x dx

Substituting in the integral: ∫ cot x log sin x dx = ∫ t dt = t²/2 + C

Substituting back t = log sin x: ∫ cot x log sin x dx = (log sin x)²/2 + C

Exercise: EXERCISE 7.2 | Q: 29 | (Page 17)
For More Understanding

Explanation

This is a standard substitution problem. The key insight is recognizing that the derivative of log sin x is cot x, which appears as a factor in the integrand. The context shows that ∫ cot x dx = log|sin x| + C, confirming that d/dx(log sin x) = cot x. This makes the substitution t = log sin x straightforward, as dt = cot x dx, simplifying the integral to ∫ t dt.

Solution Steps

  1. Step 1: Let t = log sin x

  2. Step 2: Differentiate: dt = (1/sin x)·cos x dx = cot x dx

  3. Step 3: Substitute: ∫ cot x log sin x dx = ∫ t dt

  4. Step 4: Integrate: ∫ t dt = t²/2 + C

  5. Step 5: Substitute back: = (log sin x)²/2 + C

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