Indefinite Integration
Integration by Parts with Exponential
Grade 12
Question:
<p>If <span>\(\int x^5 e^{-x^2} dx = g(x)e^{-x^2} + C\)</span>, where <span>\(C\)</span> is a constant of integration, then <span>\(g(-1)\)</span> is equal to</p>
<p>(a) <span>\(-1\)</span></p>
<p>(b) <span>\(1\)</span></p>
<p>(c) <span>\(-\frac{1}{2}\)</span></p>
<p>(d) <span>\(-\frac{5}{2}\)</span></p>
Step-by-Step Solution
Key Concept: Apply integration by parts method to separate polynomial and exponential components
<p>Use integration by parts with <span>$u = x^4$</span> and <span>$dv = xe^{-x^2}dx$</span> repeatedly to find <span>$g(x)$</span>, then evaluate at <span>$x = -1$</span>.</p>
Correct Answer: D