Find the indefinite integral $\,\int(x+8)(x+4)^{1/2}\mathrm dx\,$ by using $\,\frac{\mathrm d}{\mathrm dx}(x+4)^p\,$ where $\,p\,$ is a constant

Question

Find the indefinite integral $\displaystyle\int(x+8)(x+4)^\frac12\mathrm dx\,$ by using $\,\dfrac{\mathrm d}{\mathrm dx}(x+4)^p\,$ where $\,p\,$ is a constant.

How can I use $\,p(x+4)^{p-1}$ to solve the integral? I can't seem to find the connection between them.

Answer 1

Note that

$(x+8)(x+4)^{\frac{1}{2}}=(x+4+4)(x+4)^{\frac{1}{2}}=(x+4)(x+4)^{\frac{1}{2}}+4(x+4)^{\frac{1}{2}}$

This then becomes $$\int(x+4)^{\frac{3}{2}}+4(x+4)^{\frac{1}{2}}\mathrm{d}x$$which makes it easier to spot where to use your trick.