Question

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

• A. \(\dfrac{1}{2}\{f(x)\}^2\)
• B. \(\{f(x)\}^3\)
• C. \(\dfrac{\{f(x)\}^3}{3}\)
• D. \(\{f(x)\}^2\)

Hint:

Whenever integral looks like \(\int f(x)f'(x)\,dx\), substitute \(u=f(x)\) and use \(\int u\,du=\frac{u^2}{2}\).

Answer 1

The Correct Option is A

Step 1: Interpret the given relation.
Given:
\[ f'(x)dx = f(x) \] This implies:
\[ d(f(x)) = f'(x)dx \] So the condition is actually:
\[ d(f(x)) = f'(x)dx \] and given statement indicates substitution possible.
Step 2: Evaluate the integral.
We want:
\[ \int \{f(x)\}^2 dx \] Using substitution \(u=f(x)\).
Then:
\[ du=f'(x)dx \] Given relation supports direct integration form, so:
\[ \int u\,du = \frac{u^2}{2} \] Thus:
\[ \int \{f(x)\}^2 dx = \frac{1}{2}\{f(x)\}^2 \] Final Answer: \[ \boxed{\frac{1}{2}\{f(x)\}^2} \]