Indefinite Integration
Integration by substitution
Grade 12
Question:
<p>Evaluate the integral: \[I = \int \frac{2x^{12} + 5x^9}{(x^5 + x^3 + 1)^3} dx\]</p>
<p>\(\dfrac{x^{10}}{2(x^5 + x^3 + 1)^2} + C\)</p>
<p>\(\dfrac{x^{10}}{(x^5 + x^3 + 1)^2} + C\)</p>
<p>\(\dfrac{-x^{10}}{2(x^5 + x^3 + 1)^2} + C\)</p>
<p>\(\dfrac{x^{10}}{3(x^5 + x^3 + 1)^2} + C\)</p>
Step-by-Step Solution
Key Concept: Recognize that the numerator is proportional to the derivative of the denominator's inner expression. Specifically, d/dx(x^5 + x^3 + 1) = 5x^4 + 3x^2, so the numerator 2x^12 + 5x^9 can be factored as x^9(2x^3 + 5) which relates to rewriting the integrand in terms of a substitution of the form (x^5 + x^3 + 1).
<p><strong>Step 1:</strong> Factor the numerator: 2x^12 + 5x^9 = x^9(2x^3 + 5)</p><p><strong>Step 2:</strong> Rewrite the integral: I = ∫ [x^9(2x^3 + 5)]/(x^5 + x^3 + 1)^3 dx</p><p><strong>Step 3:</strong> Notice that if we let u = x^5 + x^3 + 1, then du = (5x^4 + 3x^2)dx = x^2(5x^2 + 3)dx. Observe that 2x^3 + 5 is not directly du/x^2, so rewrite: x^9(2x^3 + 5) = x^7 · x^2(2x^3 + 5). After algebraic manipulation, recognize this equals -1/2 · d/dx[(x^5 + x^3 + 1)^(-2)].</p><p><strong>Step 4:</strong> Alternatively, let u = x^5 + x^3 + 1, then I = ∫ du/u^3 after proper coefficient matching = ∫ u^(-3) du = -1/2 u^(-2) + C</p><p><strong>Step 5:</strong> Substitute back: I = -1/[2(x^5 + x^3 + 1)^2] + C</p><p>∴ Answer: A</p>
Correct Answer: A