Question 2 of 33beginner🔧 ApplyShort Answer2 marks

Find an anti derivative (or integral) of the following functions by the method of inspection: cos 3x

Correct Answer

To find the integral of cos⁡3x\cos 3x, we use the standard integration rule for trigonometric functions with linear arguments. The provided context demonstrates that ∫sin⁡5x dx=−15cos⁡5x\int \sin 5x \, dx = -\frac{1}{5} \cos 5x, which shows that the coefficient of xx appears in the denominator of the result.

Applying this logic to cos⁡3x\cos 3x, we treat the coefficient of xx as 33. Since the integral of cos⁡x\cos x is sin⁡x\sin x, the integral of cos⁡3x\cos 3x is obtained by dividing sin⁡3x\sin 3x by the coefficient 33. Thus, the required integral is 13sin⁡3x+C\frac{1}{3} \sin 3x + C.

Exercise: EXERCISE 7.1 | Q: 2 | (Page 10)
For More Understanding

Explanation

The question requires finding the integral of cos 3x. The textbook context provides the example of integrating sin 5x, which yields a result with the coefficient 5 in the denominator. This establishes the general rule that ∫cos(ax) dx = (1/a)sin(ax) + C. By applying this rule with a=3, the solution is derived.

Solution Steps

  1. Step 1: Identify the integration rule from the context where ∫sin⁡5x dx=−15cos⁡5x\int \sin 5x \, dx = -\frac{1}{5} \cos 5x.

  2. Step 2: Apply the corresponding rule for cosine: ∫cos⁡(ax) dx=1asin⁡(ax)+C\int \cos(ax) \, dx = \frac{1}{a} \sin(ax) + C.

  3. Step 3: Substitute a=3a = 3 to get 13sin⁡3x+C\frac{1}{3} \sin 3x + C.

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