Integration of $4xe^{-1/2}$

Question

It is given that $$\int_0^p4xe^{-\frac{1}{2}x}dx=9$$

where $p$ is a positive constant

(i) Show that $$p=2 \ln \left( \frac{8p+16}{7} \right )$$ I reached

$$8pe^{-p/2} + 16e^{-p/2} = 7$$

What are the steps to show in (i) ??

Answer 1

HINT:

So, $$(8p+16)e^{-\frac p2}=7\implies \ln(8p+16)+(-p/2)\ln(e)=\ln7$$

$\ln(e)=\log_ee=?$

Can you take it home from here using $\ln(a/b)=\ln a-\ln b$