Question

If \(\int f(x) dx = F(x)\), then \(\int x^3 f(x^2) dx\) is equal to

• A. \(\frac{1}{2}[x^2\{F(x)\}^2 - \int \{F(x)\}^2 dx]\)
• B. \(\frac{1}{2}[x^2 F(x^2) - \int F(x^2) d(x^2)]\)
• C. \(\frac{1}{2}[x^2 F(x) - \frac{1}{2}\int \{F(x)\}^2 dx]\)
• D. None of the above

Hint:

Use substitution \(t = x^2\) followed by integration by parts.

Answer 1

The Correct Option is B

Step 1: Understanding the Concept:
Substitution method: let \(t = x^2\).
Step 2: Detailed Explanation:
Let \(t = x^2\), \(dt = 2x dx\), \(x dx = dt/2\), \(x^3 dx = x^2 \cdot x dx = t \cdot (dt/2)\)
\(\int x^3 f(x^2) dx = \frac{1}{2} \int t f(t) dt\)
Integration by parts: Let \(u = t\), \(dv = f(t) dt \rightarrow du = dt\), \(v = F(t)\)
\(\int t f(t) dt = t F(t) - \int F(t) dt\)
So \(\int x^3 f(x^2) dx = \frac{1}{2}[t F(t) - \int F(t) dt] = \frac{1}{2}[x^2 F(x^2) - \int F(x^2) d(x^2)]\)
Step 3: Final Answer:
\(\frac{1}{2}[x^2 F(x^2) - \int F(x^2) d(x^2)]\).