Question

\(\int f(x)dx=g(x)+c\), then \(\int xf(x^{2})dx=\)

• A. \(2g(x^{2})+c\)
• B. \(2g(x)+c\)
• C. \(\frac{1}{2}g(x^{2})+c\)
• D. \(\frac{1}{2}g(x)+c\)

Hint:

Substitute u equal to x squared so that x dx becomes du over 2.

Answer 1

The Correct Option is C

Step 1: Differentiate the candidate
Differentiate $\frac{1}{2}g(x^{2})$: $\frac{1}{2}g'(x^{2})\cdot 2x$.

Step 2: Use g' = f
Since $g'(x)=f(x)$, this becomes $x\,f(x^{2})$.

Step 3: Conclude
It matches the integrand, so the antiderivative is correct.

Final Answer:
The answer is $\frac{1}{2}g(x^{2})+c$. \[ \boxed{\text{(C) }\dfrac{1}{2}g(x^{2})+c} \]