Answer
\[\frac{{\,{{\left( {p + 1} \right)}^7}}}{7} - \frac{{\,{{\left( {p + 1} \right)}^6}}}{6} + C\]
Work Step by Step
\[\begin{gathered}
\int_{}^{} {p\,{{\left( {p + 1} \right)}^5}dp} \hfill \\
\hfill \\
Let\,\,u = p + 1\,\,,\,\,then\,\,p = u - 1 \hfill \\
So\,\,that\,\,dp = du \hfill \\
\int_{}^{} {\,\left( {u - 1} \right){u^5}du} \hfill \\
\int_{}^{} {\,\left( {{u^6} - {u^5}} \right)du} \hfill \\
Power\,\,rule \hfill \\
\frac{{{u^{6 + 1}}}}{{6 + 1}} - \frac{{{u^{5 + 1}}}}{{5 + 1}} + C \hfill \\
\frac{{{u^7}}}{7} - \frac{{{u^6}}}{6} + C \hfill \\
Substituting\,\,u = p + 1\,\,gives \hfill \\
\frac{{\,{{\left( {p + 1} \right)}^7}}}{7} - \frac{{\,{{\left( {p + 1} \right)}^6}}}{6} + C \hfill \\
\end{gathered} \]
