Q62 / 120 · VITEEE 2010 · Mathematics

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

  • Correct answer
    A
    $\frac{1}{2}\{\mathrm{f}(\mathrm{x})\}^{2}$
  • B
    $\{f(x)\}^{3}$
  • C
    $\frac{\{f(x)\}^{3}}{3}$
  • D
    $\{f(x)\}^{2}$

Solution

$$ \begin{array}{l} \int f(x) d x=f(x) \\ \Rightarrow \frac{d}{d x} f(x)=f(x) \\ \quad\left[\because f(x)=\int \frac{d}{d x} f(x) d x\right] \end{array} $$ Now, $\int\{f(x)\}^{2} d x=\int f(x) \cdot f(x) d x$ $=f(x) \int f(x) d x-\int\left[\frac{d}{d x} f(x) \int f(x) d x\right] d x$ (integrating by parts) $$ \begin{array}{l} =f(x) f(x)-\int f(x) f(x) d x \\ \Rightarrow 2 \int\{f(x)\}^{2} d x=\{f(x)\}^{2} \\ \Rightarrow \int\{f(x)\}^{2} d x=\frac{1}{2}\{f(x)\}^{2} \end{array} $$

Hence, the correct answer is option: 'A'.

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