Question 24 of 33intermediate🔧 ApplyShort Answer2 marks

Integrate the functions in Exercises 1 to 22: x sin⁡\sin x

Correct Answer

To find the integral of x sin x, we apply the method of integration by parts.

According to the rule stated in the context, when one function is a power of x or a polynomial in x, we take it as the first function. Therefore, we take x as the first function and sin x as the second function.

Using integration by parts formula: ∫u·v dx = u∫v dx - ∫[(du/dx)·∫v dx] dx

Step 1: ∫x sin x dx = x∫sin x dx - ∫[(d(x)/dx)·∫sin x dx] dx

Step 2: From the context, ∫sin x dx = -cos x + C

Step 3: ∫x sin x dx = x(-cos x) - ∫(1)(-cos x) dx = -x cos x + ∫cos x dx

Step 4: From the context, ∫cos x dx = sin x + C

Step 5: ∫x sin x dx = -x cos x + sin x + C = sin x - x cos x + C

Exercise: EXERCISE 7.6 | Q: 1 | (Page 39)
For More Understanding

Explanation

This question tests the integration by parts technique. The context clearly states that when a polynomial in x is involved, it should be taken as the first function. The standard integrals of sin x and cos x are provided in the context, which are used to complete the solution.

Solution Steps

  1. Step 1: Identify x as first function and sin x as second function

  2. Step 2: Apply integration by parts formula

  3. Step 3: Integrate sin x to get -cos x

  4. Step 4: Integrate cos x to get sin x

  5. Step 5: Simplify to get sin x - x cos x + C

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