Question 8 of 33intermediate🔧 ApplyShort Answer2 marks

Integrate the functions in Exercises 1 to 37: sin⁡\sin x sin⁡\sin (cos⁡\cos x)

Correct Answer

Solution:

Let I = ∫ sin x sin(cos x) dx

Step 1: Put cos x = t. Then -sin x dx = dt, or sin x dx = -dt.

Step 2: Substituting in the integral: I = ∫ sin(t) · (-dt) = -∫ sin t dt

Step 3: Integrating: I = -(-cos t) + C = cos t + C

Step 4: Substituting back t = cos x: I = cos(cos x) + C

Therefore, ∫ sin x sin(cos x) dx = cos(cos x) + C

Exercise: EXERCISE 7.2 | Q: 4 | (Page 16)
For More Understanding

Explanation

This question requires the substitution method as demonstrated in the context. The context shows 'Put cos x = t so that -sin x dx = dt' for solving integrals involving sin x and functions of cos x. This substitution converts the integral into a simple form that can be integrated directly.

Solution Steps

  1. Step 1: Substitute cos x = t, so that -sin x dx = dt

  2. Step 2: Replace sin x dx with -dt in the integral

  3. Step 3: Integrate -∫ sin t dt to get cos t + C

  4. Step 4: Substitute back t = cos x to get cos(cos x) + C

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