Answer
$$\int\frac{dx}{\sqrt{5x+8}}=\frac{2\sqrt{5x+8}}{5}+C$$
Work Step by Step
$$A=\int\frac{dx}{\sqrt{5x+8}}$$
a) We set $u=5x+8$.
Then $$du=5dx$$
That means, $$dx=\frac{1}{5}du$$
Therefore, $$A=\frac{1}{5}\int\frac{du}{\sqrt u}=\frac{1}{5}\int u^{-1/2}du$$ $$A=\frac{1}{5}\times\frac{u^{1/2}}{\frac{1}{2}}+C=\frac{1}{5}\times2\sqrt u+C=\frac{2\sqrt u}{5}+C$$ $$A=\frac{2\sqrt{5x+8}}{5}+C$$
b) We set $u=\sqrt{5x+8}$.
Then $$du=\frac{(5x+8)'}{2\sqrt{5x+8}}dx=\frac{5}{2u}dx$$
That means, $$dx=\frac{2u}{5}du$$
Therefore, $$A=\int\frac{\frac{2u}{5}}{u}du=\int\frac{2u}{5u}du=\int\frac{2}{5}du$$ $$A=\frac{2u}{5}+C$$ $$A=\frac{2\sqrt{5x+8}}{5}+C$$