Exponential distribution $P(Z \geq 5)$ Exponential distribution 
Let $Z ∼ Exponential(4)$. Compute each of the following
(a) $P(Z \geq 5)$
$$P(Z \geq 5) = \int_{5}^{\infty} 4e^{-4x}dx$$
Let $u = -4x$, then $du = -4dx \leftrightarrow -\frac{1}{4}du = dx$
$$-\int_{-\infty}^{-20} e^{u} du =  -e^{u}|_{-\infty}^{-20} = -(e^{-20} - \lim_{u\to-\infty}e^{A}) = -e^{-20} + 0 = -e^{-20}$$
Answer is $e^{-20}$. Where did I go wrong or is the solution wrong?

Just this:
$$\int_5^\infty 4 e^{-4x}\mathsf d x= \lim_{x\to\infty}(-e^{-4x})-(-e^{-4\cdot 5})$$
When you apply the substitution
$$\begin{split}\int_{-20}^{-\infty} 4e^{u}\dfrac{\mathsf d u}{-4} &= -\int_{-20}^{-\infty}e^u\mathsf d u \\ &=\int_{-\infty}^{-20}e^u\mathsf d u \\ &=e^{-20}-\lim_{u\to-\infty}e^u\end{split}$$
