Integrate $s(x)= e^{-0.014x^{0.88}}$ I have a survival function $s(x)= e^{-0.014x^{0.88}}$
Now, I wanna find the surface under this curve at $x=1015$. However, I get stuck as I get a negative outcome. Could anyone help me out? 
 A: You can write your integral as 
\begin{equation}
 {\displaystyle\int}\mathrm{e}^{-\frac{7x^\frac{22}{25}}{500}}\,\mathrm{d}x
\end{equation}
Take the following change of variable 
\begin{equation}
 u=\dfrac{7^\frac{25}{22}x}{2^\frac{25}{11}{\cdot}125^\frac{25}{22}}
\end{equation}
which means that $\mathrm{d}x=\dfrac{2^\frac{25}{11}{\cdot}125^\frac{25}{22}}{7^\frac{25}{22}}\,\mathrm{d}u$. You get
\begin{equation}
 {\displaystyle\int}\mathrm{e}^{-\frac{7x^\frac{22}{25}}{500}}\,\mathrm{d}x
 ={{\dfrac{2^\frac{25}{11}{\cdot}125^\frac{25}{22}}{7^\frac{25}{22}}}}{\displaystyle\int}\mathrm{e}^{-u^\frac{22}{25}}\,\mathrm{d}u
\end{equation}
The integral ${\displaystyle\int}\mathrm{e}^{-u^\frac{22}{25}}\,\mathrm{d}u$ is known as the Incomplete Gamma Function, i.e.
\begin{equation}
 {\displaystyle\int}\mathrm{e}^{-u^\frac{22}{25}}\,\mathrm{d}u
=-\dfrac{25\operatorname{\Gamma}\left(\frac{25}{22},u^\frac{22}{25}\right)}{22}
\end{equation}
Replacing and undoing the change of variable, we get
\begin{equation}
 {\displaystyle\int}\mathrm{e}^{-\frac{7x^\frac{22}{25}}{500}}\,\mathrm{d}x
 =-\dfrac{25{\cdot}2^\frac{14}{11}{\cdot}125^\frac{25}{22}\operatorname{\Gamma}\left(\frac{25}{22},\frac{7x^\frac{22}{25}}{500}\right)\sqrt[25]{x}}{11{\cdot}7^\frac{25}{22}\left|\sqrt[25]{x}\right|}+C
\end{equation}
