find the indefinite integral. (remember the constant of integration.) ∫e^(-0.2x) dx

Explanation:

Step1: Use substitution method

Let $u=- 0.2x$, then $du=-0.2dx$, and $dx=-\frac{1}{0.2}du = - 5du$.

Step2: Rewrite the integral

The integral $\int e^{-0.2x}dx$ becomes $\int e^{u}\times(-5du)=-5\int e^{u}du$.

Step3: Integrate $e^{u}$

We know that $\int e^{u}du = e^{u}+C$. So $-5\int e^{u}du=-5e^{u}+C$.

Step4: Substitute back $u$

Substitute $u = - 0.2x$ back, we get $-5e^{-0.2x}+C$.

Answer:

$-5e^{-0.2x}+C$