How can I solve this?
First,
$\displaystyle p(x)\geq 0$.
Next we need to find,
$\displaystyle \int_{-\infty}^{\infty} p(x)dx$
Subdivide the interval as,
$\displaystyle \int_{-\infty}^0p(x)dx+\int_0^{\infty}p(x)dx$
Since the integral is continous accept at possibly one point we can chose to ignore that fact.
Now,
$\displaystyle \int_{-\infty}^0 p(x)dx=0$ because $\displaystyle p(x)=0$ for $\displaystyle x<0$.
And,
$\displaystyle \int_0^{\infty} p(x)dx=\int_0^{\infty} \lambda e^{-\lambda x}dx$
The anti-derivative is,
$\displaystyle e^{-\lambda x} | ^{\infty}_0 =1$
So it is indeed a probability density function.