Question

If $\int f(x)dx=F(x)$, then $\int x³f(x²)dx$ 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:

If $\int f(x)dx=F(x)$, then $\int x)dx$ is equal to

Answer 1

The Correct Option is B

Step 1: Concept
Use the substitution method and integration by parts.
Step 2: Analysis
Let $x^{2} = t$, then $2x\,dx = dt$. The integral becomes $\frac{1}{2} \int t \cdot f(t) \, dt$.
Step 3: Evaluation
Apply integration by parts on $\int t \cdot f(t) \, dt$:
$\int t \cdot f(t) \, dt = t \cdot F(t) - \int F(t) \, dt$.
Step 4: Conclusion
Substituting back $t = x^2$ gives $\frac{1}{2} [x^2 F(x^2) - \int F(x^2) \, d(x^2)]$.
Final Answer: (b)