Integral $\int_0^\infty |x-c|e^{-2x}dx$

Question

I have to evaluate the integral: $$\int_0^\infty |x-c|e^{-2x}dx$$ with c $\in \mathbb{R}$.

I would evaluate the integral this way: http://math.ucr.edu/~jmd/9B_S14_AbsInt.pdf. This would give me one solution. However, the provided result to this problem divides the problem to: if c negative or positive. (c negative: $\frac{1}{4} - \frac{c}{2}$, c positive: $\frac{1}{2}e^{-2c}-\frac{1}{4}+\frac{c}{2}$).

How should this integral be solved to get the mentioned result?

Answer 1

Hint: Write the given integral on this manner if $c>0$

$$\int_0^\infty=\int_0^c+\int_c^\infty$$ and integrate by parts and notice that if $c<0$ we have $|x-c|=x-c$ for $x\ge0$.