Question

50 points each question. Please help. How do I solve?

Answer 1

`I=\displaystyle \int (\sqrt x)/(\sqrt x -4) ~dx\\\\\text{Let,}\\\\~~~~u=\sqrt x-4\\\\\implies (du)/(dx) = \frac d{dx} \left( \sqrt x -4 \right)\\\\\implies (du)/(dx) = \frac 1{2 \sqrt x} \\\\\implies dx =2\sqrt x~ du\\\\\\I= \displaystyle \int (u+4)/(u+4 -4 ) 2(u+4) du\\\\\\~~~=2\displaystyle \int ((u+4)^2)/(u) du\\\\\\~~~=2\displaystyle \int (u^2+ 8u+16)/(u) du\\\\\\~~~=2\displaystyle \int \left(u+8+\frac{16}u\right) du\\\\\\`

`~~~=2\left(\displaystyle \int u ~ du + \displaystyle \int 8 ~ du + \displaystyle \int \frac{16} u ~ du\right) \\\\\\~~~=2\left( \frac {u^2}2 + 8u+16\ln |u| \right) +C\\\\\\~~~~= u^2 + 16u+ 32 \ln |u|+C\\\\\\~~~~=\left( \sqrt x -4 \right)^2 + 16\left( \sqrt x -4 \right) +32 \ln |\sqrt x -4|+C\\\\\\~~~~= x-8\sqrt x+16 +16\sqrt x -64 +32 \ln \left|\sqrt x-4 \right|+C\\\\\\\~~~~=x+8\sqrt x -48 +32\ln|\sqrt x -4| +C`