I have a density function defined as $f(x) = (1/10) \exp(-x/10)$. The answer to the solution suggests that the antiderivative, the cumulative distribution function, ought to be $\exp(-x/10)$, however my own attempts at arriving at the solution results in $(1/10) \exp(-x/10)$ as the solution.
What am I not taking into consideration?
We have the integral (antiderivative) $$ \int \frac{1}{10} e^{-\frac{1}{10}x} dx $$ If we set $u=-\frac{1}{10}x$, then we need to change $dx$ to its equivalent form for our variable $u$. Here we can write $$ \frac{du}{dx} = \frac{d}{dx} \left( -\frac{1}{10}x \right) = -\frac{1}{10} $$ From here we can say that $$ dx = -10du $$ So we can substitute this into out integral to write $$ \int \frac{1}{10} e^u (-10du) = \int -e^u du = -\int e^u du = -e^u + C $$ We substitute back in the expression for $u$ to get $$ -e^{-\frac{1}{10}x} + C $$