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}

Advertisements

Leave a Reply

Fill in your details below or click an icon to log in:

WordPress.com Logo

You are commenting using your WordPress.com account. Log Out / Change )

Twitter picture

You are commenting using your Twitter account. Log Out / Change )

Facebook photo

You are commenting using your Facebook account. Log Out / Change )

Google+ photo

You are commenting using your Google+ account. Log Out / Change )

Connecting to %s