# integration under integral sign

Let

 $I(\alpha)\;=\;\int_{a}^{b}\!f(x,\,\alpha)\,dx.$

where  $f(x,\,\alpha)$ is continuous in the rectangle

 $a\leqq x\leqq b,\,\quad\alpha_{1}\leqq\alpha\leqq\alpha_{2}.$

Then  $\alpha\mapsto I(\alpha)$  is continuous and hence integrable (http://planetmath.org/RiemannIntegrable) on the interval$\alpha_{1}\leqq\alpha\leqq\alpha_{2}$;  we have

 $\int_{\alpha_{1}}^{\alpha_{2}}I(\alpha)\,d\alpha\;=\;\int_{\alpha_{1}}^{\alpha% _{2}}\left(\int_{a}^{b}\!f(x,\,\alpha)\,dx\right)d\alpha.$

This is a double integral over a in the $x\alpha$-plane, whence one can change the order of integration (http://planetmath.org/FubinisTheorem) and accordingly write

 $\int_{\alpha_{1}}^{\alpha_{2}}\left(\int_{a}^{b}\!f(x,\,\alpha)\,dx\right)d% \alpha\;=\;\int_{a}^{b}\left(\int_{\alpha_{1}}^{\alpha_{2}}\!f(x,\,\alpha)\,d% \alpha\right)dx.$

Thus, a definite integral depending on a parametre may be integrated with respect to this parametre by performing the integration under the integral sign.

Example.  For being able to evaluate the improper integral

 $I\;=\;\int_{0}^{\infty}\frac{e^{-ax}-e^{-bx}}{x}\,dx\qquad(a>0,\;b>0),$

we may interprete the integrand as a definite integral:

 $\frac{e^{-ax}-e^{-bx}}{x}\;=\;\operatornamewithlimits{\Big{/}}_{\!\!\!\alpha=b% }^{\,\quad a}\!\frac{e^{-\alpha x}}{x}\;=\;\int_{a}^{b}\!e^{-\alpha x}\,d\alpha.$

Accordingly, we can calculate as follows:

 $\displaystyle I$ $\displaystyle\;=\;\int_{0}^{\infty}\left(\int_{a}^{b}\!e^{-\alpha x}\,d\alpha% \right)dx$ $\displaystyle\;=\;\int_{a}^{b}\left(\int_{0}^{\infty}\!e^{-\alpha x}\,dx\right% )d\alpha$ $\displaystyle\;=\;\int_{a}^{b}\left(\operatornamewithlimits{\Big{/}}_{\!\!\!x=% 0}^{\,\quad\infty}\!-\frac{e^{-\alpha x}}{\alpha}\right)d\alpha$ $\displaystyle\;=\;\int_{a}^{b}\!\frac{1}{\alpha}\,d\alpha\;=\;% \operatornamewithlimits{\Big{/}}_{\!\!\!a}^{\,\quad b}\!\ln{\alpha}$ $\displaystyle\;=\;\ln\frac{b}{a}$
Title integration under integral sign IntegrationUnderIntegralSign 2013-03-22 18:46:27 2013-03-22 18:46:27 pahio (2872) pahio (2872) 5 pahio (2872) Theorem msc 26A42 FubinisTheorem DifferentiationUnderIntegralSign RelativeOfExponentialIntegral