Question

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

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Answer 1

We want to find the integral of the function $e^{3x}$.We know the integral of $e^{x}is{e}^x$ itself. We can use this result along with theorems on integration to solve this problem. It says $\int f(ax+b)=\frac{F(ax+b)}{a}+C$

We have $f\left(x\right)=e^{x}andF\left(x\right)=e^x$.

Now we want to find integral of f(ax+b) or $e^{3x}$

$\Rightarrow a=3andb=0$

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

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

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

$\Rightarrow c=0$

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

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