If $\int \frac{\mathrm{dt}}{(1+\sqrt{\mathrm{t}}{)}^8}=\frac{-1}{3(1+\sqrt{\mathrm{t}}{)}^{{\mathrm{p}}_1}}+\frac{2}{7(1+\sqrt{\mathrm{t}}{)}^{{\mathrm{p}}_2}}+\mathrm{C}$ Where C is constant of integration, then

$1+\sqrt{\mathrm{t}}=\mathrm{x}\Rightarrow \sqrt{\mathrm{t}}=\mathrm{x}-1$

$\frac{1}{2\sqrt{\mathrm{t}}}\mathrm{dt}=\mathrm{dx}\Rightarrow \mathrm{dt}=2\sqrt{\mathrm{t}}\mathrm{dx}$

                    $\mathrm{dt}=2(\mathrm{x}-1)\mathrm{dx}$

$\int \frac{2(\mathrm{x}-1)\mathrm{dx}}{{\mathrm{x}}^8}=2\int \left[\frac{1}{{\mathrm{x}}^7}-\frac{1}{{\mathrm{x}}^8}\right]\mathrm{dx}$

                    $\begin{array}{l}=2\left[\int {\mathrm{x}}^{-7}\mathrm{dx}-\int {\mathrm{x}}^{-8}\mathrm{dx}\right]\\ =2\left[\frac{1}{{\mathrm{x}}^6(-6)}-\frac{1}{{\mathrm{x}}^7(-7)}\right]+\mathrm{C}\\ =2\left[\frac{1}{7{\mathrm{x}}^7}-\frac{1}{6{\mathrm{x}}^6}\right]+\mathrm{C}\\ =2\left[\frac{1}{7(1+\sqrt{\mathrm{t}}{)}^7}-\frac{1}{6(1+\sqrt{\mathrm{t}}{)}^6}\right]+\mathrm{C}\end{array}$

$\begin{array}{l}=\frac{2}{7(1+\sqrt{\mathrm{t}}{)}^7}-\frac{1}{3(1+\sqrt{\mathrm{t}}{)}^6}+\mathrm{c}\\ \therefore {{\mathrm{p}}_2}^{ }=7,{\mathrm{p}}_1=6\end{array}$