Indefinite Integration
Integration with Exponential Functions
Grade 12
Question:
<p>If <span>\(\int 4x^3 e^{-4x^3} dx = \frac{1}{48}e^{-4x^3}f(x) + C\)</span>, where <span>\(C\)</span> is a constant of integration, then <span>\(f(x)\)</span> is equal to</p>
<p>(a) <span>\(-4x^3 - 1\)</span></p>
<p>(b) <span>\(4x^3 + 1\)</span></p>
<p>(c) <span>\(-4x^3 - 1\)</span></p>
<p>(d) [Option not clearly visible]</p>
Step-by-Step Solution
Key Concept: Apply substitution to simplify the integral, then identify the polynomial factor <span>\(f(x)\)</span>
<p>Use substitution <span>\(u = -4x^3\)</span> and integration by parts to obtain the form with <span>\(f(x)\)</span>.</p>
Correct Answer: A