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: Understanding the Concept:
Integration by parts: \(\int u\,dv=uv-\int v\,du\). Applied with \(u=\int f\,dx\) and \(v=\int g\,dx\) gives a formula for the product of two integrals.

Step 2: Differentiate the product:
Let \(F=\int f\,dx\) and \(G=\int g\,dx\). Then
\[ \frac{d}{dx}(FG)=F'G+FG'=fG+gF \]

Step 3: Integrate back:
\[ FG=\int\left[f(x)\int g(x)dx+g(x)\int f(x)dx\right]dx \]

Step 4: Choose:
This is option (C). Options (A) and (B) put the derivatives \(f'\) and \(g'\) in, which is not the rule. Option (D) has a minus sign, which would be wrong because the product rule adds.

Final Answer:
The product rule gives option (C). \[ \boxed{\int\left[f\int g\,dx+g\int f\,dx\right]dx} \]