Compute
![\[ \int_{-2}^{-4} (x+4)^{10} \, dx. \]](https://www.stumblingrobot.com/wp-content/ql-cache/quicklatex.com-1712c651dd3c00a246201c4f00c71b73_l3.png "Rendered by QuickLaTeX.com")
We really don’t want to expand 
so we use the invariance of the integral under translation (Theorem 1.18 in Apostol) to make things easier
![\[ \int_{-2}^{-4} (x+4)^{10} \, dx = \int_2^0 x^{10} \, dx = - \int_0^2 x^{10} \, dx = \left. -\frac{x^{11}}{11} \right|_0^2 = - \frac{2^{11}}{11}. \]](https://www.stumblingrobot.com/wp-content/ql-cache/quicklatex.com-d4ee1c3db924ab135cb4d1de22d5782e_l3.png "Rendered by QuickLaTeX.com")
Compute
We really don’t want to expand so we use the invariance of the integral under translation (Theorem 1.18 in Apostol) to make things easier