Indefinite Integration
Integration by Parts
Grade 12
Question:
<p>If \(\int f(x)dx = \Psi(x)\), then \(\int x^5 f(x^3)\,dx\) is equal to</p>
<p>\(\dfrac{1}{3}x^3\Psi(x^3) - 3\int x^3\Psi(x^3)\,dx + C\)</p>
<p>\(\dfrac{1}{3}x^3\Psi(x^3) - \int x^2\Psi(x^3)\,dx + C\)</p>
<p>\(\dfrac{1}{3}\left[x^3\Psi(x^3) - \int x^3\Psi(x^3)\,dx\right] + C\)</p>
<p>\(\dfrac{1}{3}\left[x^3\Psi(x^3) - \int x^2\Psi(x^3)\,dx\right] + C\)</p>
Step-by-Step Solution
Key Concept: Recognize that x^5·f(x³) requires substitution u = x³, which transforms the integrand into a form directly involving the given antiderivative Ψ(x). The key is matching the differential: du = 3x²dx, so x⁵dx = (x³/3)du.
<p><strong>Step 1:</strong> Given ∫f(x)dx = Ψ(x), we need to find ∫x⁵f(x³)dx</p><p><strong>Step 2:</strong> Use substitution u = x³, so du = 3x²dx, which means x²dx = du/3</p><p><strong>Step 3:</strong> Rewrite x⁵f(x³)dx = x³·x²·f(x³)dx = u·f(u)·(du/3)</p><p><strong>Step 4:</strong> Therefore: ∫x⁵f(x³)dx = ∫u·f(u)·(du/3) = (1/3)∫u·f(u)du</p><p><strong>Step 5:</strong> By the given condition, ∫f(u)du = Ψ(u), so differentiating: Ψ'(u) = f(u)</p><p><strong>Step 6:</strong> We need ∫u·f(u)du. Using integration by parts or recognizing the pattern: if d/du[u·Ψ(u)] = Ψ(u) + u·Ψ'(u) = Ψ(u) + u·f(u), then ∫u·f(u)du = u·Ψ(u) - ∫Ψ(u)du</p><p><strong>Step 7:</strong> Substituting back u = x³: ∫x⁵f(x³)dx = (1/3)[x³·Ψ(x³) - ∫Ψ(x³)dx] = <strong>(x³/3)Ψ(x³) + C</strong></p><p>∴ Answer: <strong>(x³/3)Ψ(x³)</strong></p>
Correct Answer: D