Question 4 of 33beginner🔧 ApplyShort Answer2 marks

Find the following integrals in Exercises 6 to 20: ∫\int (ax2ax^{2} + bx + c) , dx

Correct Answer

To evaluate ∫(ax² + bx + c) dx, we apply the standard integration formula for polynomials term by term.

Using the power rule of integration, ∫xⁿ dx = xⁿ⁺¹/(n+1) + C, we integrate each term separately.

∫(ax² + bx + c) dx = a·x³/3 + b·x²/2 + c·x + C

= (ax³)/3 + (bx²)/2 + cx + C

This is the required integral where C is the constant of integration.

Exercise: EXERCISE 7.1 | Q: 8 | (Page 10)
For More Understanding

Explanation

The question asks for the integral of a quadratic polynomial ax² + bx + c. This is a direct application of the power rule for integration. The context discusses various integration techniques for rational functions and expressions involving ax² + bx + c in denominators, but for integrating the polynomial itself, students apply the fundamental power rule: ∫xⁿ dx = xⁿ⁺¹/(n+1) + C. Each term is integrated separately using the linearity property of integrals.

Solution Steps

  1. Step 1: Apply the power rule ∫xⁿ dx = xⁿ⁺¹/(n+1) + C to each term

  2. Step 2: ∫ax² dx = a·x³/3

  3. Step 3: ∫bx dx = b·x²/2

  4. Step 4: ∫c dx = cx

  5. Step 5: Combine all terms: (ax³)/3 + (bx²)/2 + cx + C

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