Indefinite Integration
Integration by Parts
Grade 12
Question:
<p>85. If \(\int f(x) dx = F(x)\), then \(\int x^3 f(x^2) dx\) is equal to</p>
<p>(a) \(\frac{1}{2}\left\{x^2 F(x^2) - \int F(x^2) dx^2\right\}\)</p>
<p>(b) \(\frac{1}{2}\left\{x^2 F(x^2) - \int F(x^2) dx\right\}\)</p>
<p>(c) \(\frac{1}{2}\left\{x^2 F(x^2) - \int F(x) dx\right\}\)</p>
<p>(d) None of these</p>
Step-by-Step Solution
Key Concept: Use substitution $u = x^2$ to convert the integral, then apply integration by parts with $F(u)$.
<p>Solution not provided in the source text.</p>
Correct Answer: B