Question

use substitution to find the indefinite integral.

int p ( p + 6 ) ^ { 5 } d p

int p ( p + 6 ) ^ { 5 } d p =

Explanation:

Step1: Let \( u = p + 6 \), then \( p=u - 6 \) and \( dp=du \)

Substitute into the integral: \(\int p(p + 6)^{5}dp=\int(u - 6)u^{5}du\)

Step2: Expand the integrand

\(\int(u - 6)u^{5}du=\int(u^{6}-6u^{5})du\)

Step3: Integrate term - by - term

Using the power rule \(\int x^{n}dx=\frac{x^{n + 1}}{n+1}+C(n
eq - 1)\), we have \(\int u^{6}du-\int6u^{5}du=\frac{u^{7}}{7}-6\times\frac{u^{6}}{6}+C\)

Step4: Simplify and substitute back \( u = p + 6 \)

\(\frac{u^{7}}{7}-u^{6}+C=\frac{(p + 6)^{7}}{7}-(p + 6)^{6}+C\)

Answer:

\(\frac{(p + 6)^{7}}{7}-(p + 6)^{6}+C\)