Question

If the integral of the function $e^{3x}$ is g(x) and $g\left(0\right)=\frac{1}{3}$, then the value of $\frac{3}{e}g\left(\frac{1}{3}\right)$ is

• A. 1
• B. 3
• C. e

Answer 1

The correct option is A 1

We know that $\int e^{x}dx=e^x+c$ and $\int f(ax+b)dx=\frac{F(ax+b)}{a}+C$

We have $f\left(x\right)=e^x$ and $F\left(x\right)=e^x$.

If $f(ax+b)=e^{3x}$, $a=3andb=0$

$\Rightarrow \int f(ax+b)dx=\int e^{3x}.dx$

$=\frac{e^{3x}}{3}+c=g\left(x\right)$

We are given $g\left(0\right)=\frac{1}{3}$

$\Rightarrow c=0$

Now, $3g\left(\frac{1}{3}\right)=e$

Thus, $\frac{3}{e}g\left(\frac{1}{3}\right)$
$=\frac{1}{e}e$
$=1$