Question

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

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

Answer 1

Answer: (A)

Since, $\int f \left(x\right) dx = f \left(x\right) $
$\Rightarrow \frac{d}{dx} f\left(x\right) =f\left(x\right) $
$\left[\because f\left(x\right) =\int \frac{d}{dx} f\left(x\right)dx\right] $
Now, $ \int \left\{f\left(x\right)\right\}^{2} dx =\int f\left(x\right). f\left(x\right)dx $
$=f\left(x\right)\int f\left(x\right)dx -\int \left[\frac{d}{dx} f\left(x\right) \int f\left(x\right)dx\right]dx $
(integrating by parts)
$=f\left(x\right)f\left(x\right)-\int f\left(x\right)f\left(x\right)dx $
$\Rightarrow 2 \int\left\{f\left(x\right)\right\}^{2} dx = \left\{f\left(x\right)\right\}^{2} $
$\Rightarrow \int \left\{f\left(x\right)\right\}^{2}dx = \frac{1}{2} \left\{f\left(x\right)\right\}^{2}$
Alternate
$\int f(x) d x=f(x)$
$\left[\because \int \frac{d}{d x} f(x) d x=f(x)\right]$
$f(x)=\frac{d}{d x} f(x)$
$\Rightarrow \int d x=\int \frac{d\{f(x)\}}{f(x)},$ on integrating
$\log c +x =\log f(x)$
$f(x) =c e^{x}$ ...(i)
$\{f(x)\}^{2} =c^{2} e^{2 x}$
On integrating, $\int\{f(x)\}^{2} d x=\frac{c^{2}}{2} e^{2 x}$, [From Eq. (i)]
$\int\{f(x)\}^{2} d x=\frac{1}{2}\left\{c e^{x}\right\}^{2}=\frac{1}{2}\{f(x)\}^{2}$