Indefinite Integration
Integration by parts
Grade 12
Question:
<p>If \(\int f(x)\,dx = \Psi(x)\), then \(\int x^5 f(x^3)\,dx\) equals:</p>
<p>\(\dfrac{1}{3}x^3\Psi(x^3) - \int x^2 \Psi(x^3)\,dx + C\)</p>
<p>\(\dfrac{1}{3}\left[x^3\Psi(x^3) - \int x^2\Psi(x^3)\,dx\right] + C\)</p>
<p>\(\dfrac{1}{3}x^3\Psi(x^3) + \int x^2\Psi(x^3)\,dx + C\)</p>
<p>\(\dfrac{1}{3}\left[x^3\Psi(x^3) + \int x^2\Psi(x^3)\,dx\right] + C\)</p>
Step-by-Step Solution
Key Concept: Recognize that x^5·f(x³) can be rewritten as x²·(x³)·f(x³), enabling a substitution u = x³ that converts the integrand into a form involving Ψ(u). The key is identifying that du = 3x²dx, so x²dx = du/3.
<p><strong>Step 1:</strong> Rewrite the integral strategically.</p><p>∫x⁵f(x³)dx = ∫x²·x³·f(x³)dx</p><p><strong>Step 2:</strong> Apply substitution u = x³.</p><p>Then du = 3x²dx, so x²dx = du/3</p><p><strong>Step 3:</strong> Transform the integral.</p><p>∫x²·x³·f(x³)dx = ∫u·f(u)·(du/3) = (1/3)∫u·f(u)du</p><p><strong>Step 4:</strong> Recognize that ∫f(x)dx = Ψ(x), so ∫u·f(u)du requires integration by parts or the relation gives us a form involving Ψ.</p><p>By integration by parts: ∫u·f(u)du = u·Ψ(u) - ∫Ψ(u)du</p><p><strong>Step 5:</strong> Substitute back u = x³.</p><p>∴ ∫x⁵f(x³)dx = (1/3)[x³·Ψ(x³) - ∫Ψ(x³)dx] + C</p><p>Or simplified form: <strong>(1/3)x³Ψ(x³) + C</strong></p>
Correct Answer: B