Definite Integration
Integration by Substitution
Grade 12
Question:
<p>The value of the definite integral \[\int_0^{10} \left((x-5) + (x-5)^2 + (x-5)^3\right) dx\]</p>
<p>(a) \(\frac{125}{3}\)</p>
<p>(b) \(\frac{250}{3}\)</p>
<p>(c) \(\frac{125}{6}\)</p>
<p>(d) \(\frac{250}{4}\)</p>
Step-by-Step Solution
Key Concept: Use substitution and symmetry properties; odd functions integrate to zero over symmetric intervals.
<p><strong>Solution:</strong> Let $u = x - 5$, then $du = dx$. When $x = 0$, $u = -5$; when $x = 10$, $u = 5$.</p><p>$$\int_{-5}^{5} (u + u^2 + u^3) du = \left[\frac{u^2}{2} + \frac{u^3}{3} + \frac{u^4}{4}\right]_{-5}^{5}$$</p><p>The odd function parts $u$ and $u^3$ vanish. Only $u^2$ and $u^4$ contribute from both bounds:</p><p>$$2\left[\frac{25}{2} + \frac{625}{4}\right] = 2\left(\frac{50 + 625}{4}\right) = 2 \cdot \frac{675}{4} = \frac{250}{3}$$</p>
Correct Answer: b