Showing that $\int_0^\infty$ $\frac{1}{\theta}e^{-x/\theta}$ $dx = 1$

Question

Show the pdf is a valid pdf of an exponential random variable by showing that

$\int_0^\infty$ $\frac{1}{\theta}e^\frac{-x}{\theta}$ $dx = 1$

Answer 1

One may recall that $$ \int e^{\lambda x}dx= \frac{e^{\lambda x}}{\lambda},\quad \lambda \neq0 $$ giving $$ \int\frac{1}{\theta}e^{\large\frac{-x}{\theta}}\:dx=\frac{1}{\theta}\int e^{\large-\frac{x}{\theta}}\:dx=? $$