Question

How to find integrals in this example:
(intergral) 25 * (e^-0.2t) * dt

Answer 1

A u-substitution would work.
Let u = -0.2 t.
So du = -0.2 dt. Or [1/(-0.2)] du = dt.
Or -5 du = dt.
So our original integral is equivalent to
(integral) 25 * e^u * (-5) du.

integral e^u du is just e^u + C.
So (integral) 25 * e^u * (-5) du = -125 e^u + C.

Since u = -0.2 t, we can express answer in terms of t,
and we get -125 e^(-0.2 t) + C as our final answer.