Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

What are the steps to calculate the value of c in the following integral equation? $$ \int_{0}^{\infty}\int_{0}^{\infty}\int_{0}^{\infty}c.e^{-(x_1+2x_2+3x_3)}\,dx_1 \, dx_2 \, dx_3 = 1 $$

share|cite|improve this question
up vote 4 down vote accepted

Since $e^{-x_1+2x_2+3x_3}=e^{-x_1}e^{-2x_2}e^{-3x_3}$, we have $$ \int_0^\infty\int_0^\infty\int_0^\infty e^{-x_1+2x_2+3x_3}=\left(\int_0^\infty e^{-x_1}\right)\left(\int_0^\infty e^{-2x_2}\right)\left(\int_0^\infty e^{-3x_3}\right)=1\,\frac12\,\frac13=\frac16. $$

So $c=6$.

share|cite|improve this answer

The nice thing here is that you can rewrite this as $$c\int_0^\infty e^{-3x_3}\int_0^\infty e^{-2x_2}\int_0^\infty e^{-x_1}\,dx_1\,dx_2\,dx_3=1.$$ Each of the integrals involved is simple to evaluate, and from there, $c$ falls right out.

share|cite|improve this answer

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.