Question

If \(f(x)\) and \(g(x)\) are integrable functions then \([\int f(x)dx][\int g(x)dx] =\)

• A. \(\int [f(x)g^'(x)+f^'(x)g(x)]dx\)
• B. \(\int [f(x)g^'(x)-f^'(x)g(x)]dx\)
• C. \(\int [f(x)\int g(x)dx+g(x)\int f(x)dx]dx\)
• D. \(\int [f(x)\int g(x)dx-g(x)\int f(x)dx]dx\)

Hint:

Differentiate the product of the two integrals.

Answer 1

The Correct Option is C

Step 1: Test with simple functions:
Take $f=1$ and $g=1$. The left side is $x\cdot x=x^2$.

Step 2: Test option (C):
$\int[1\cdot x+1\cdot x]dx=\int2x\,dx=x^2$. Correct.

Step 3: Test option (D):
$\int[x-x]dx=0$. Wrong. Options (A) and (B) give $\int0\,dx$ or $\int0$ as well, since $f'=g'=0$, so also wrong.

Final Answer:
Only option (C) reproduces x squared. \[ \boxed{C} \]