Question 3 of 33beginner🔧 ApplyShort Answer2 marks

Find an anti derivative (or integral) of the following functions by the method of inspection: (ax + b)^2

Correct Answer

∫(ax + b)² dx = (ax + b)³/(3a) + C

Alternatively: ∫(ax + b)² dx = a²x³/3 + abx² + b²x + C

Exercise: EXERCISE 7.1 | Q: 4 | (Page 10)
For More Understanding

Explanation

This question asks to integrate the function (ax + b)². The context shows integration techniques including substitution and expansion methods. For this standard integral, substitution is the most efficient approach: let u = ax + b, then du = a dx, giving dx = du/a. The integral becomes (1/a)∫u² du = u³/(3a) + C = (ax + b)³/(3a) + C. Alternatively, expanding (ax + b)² = a²x² + 2abx + b² and integrating term by term gives the equivalent result.

Solution Steps

  1. Step 1: Let u = ax + b, then du/dx = a, so dx = du/a

  2. Step 2: Substitute: ∫(ax + b)² dx = ∫u² · (du/a) = (1/a)∫u² du

  3. Step 3: Integrate: (1/a) · u³/3 + C = u³/(3a) + C

  4. Step 4: Substitute back: (ax + b)³/(3a) + C

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