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