Posted in Calculus

## An interesting integral: $latex \int_0^\infty e^{-6t}dt$

An interesting integral: $\int_0^\infty e^{-6t}dt$.

This is the problem on Mathematics of Stackexchange.com.

The solution of Dror helps us learn the following trick: $de^y = e^y.dy \Rightarrow dy = \dfrac{de^y}{e^y}$. So $\int_{-\infty}^xe^{y}dy=\int_{-\infty}^xde^{y}=e^x- \lim_{y \to -\infty}e^y=e^x-0=e^x$.

Dror’s solution:
So, by substituting $y=-6t$, we can use this propety. We calculate:
$dt=-\frac{1}{6}dy, 0 \to0,\infty \to - \infty$

The integral then becomes:
$\int_0^\infty e^{-6t}dt=\int_0^{-\infty}-\frac{1}{6}e^{y}dy=\frac{1}{6}\int_{-\infty}^0e^{y}dy=\frac{1}{6}*e^0=\frac{1}{6}*1=\frac{1}{6}$