Question

1. $\int \frac{dx}{\sqrt{5x + 8}}$

Explanation:

Step1: Use substitution

Let \(u = 5x+8\), then \(du=5dx\), and \(dx=\frac{1}{5}du\).
The integral becomes \(\int\frac{1}{\sqrt{u}}\cdot\frac{1}{5}du=\frac{1}{5}\int u^{-\frac{1}{2}}du\).

Step2: Integrate using power rule

The power rule for integration is \(\int x^{n}dx=\frac{x^{n + 1}}{n+1}+C\) (\(n
eq - 1\)).
For \(n=-\frac{1}{2}\), \(\frac{1}{5}\int u^{-\frac{1}{2}}du=\frac{1}{5}\cdot\frac{u^{-\frac{1}{2}+1}}{-\frac{1}{2}+1}+C=\frac{1}{5}\cdot\frac{u^{\frac{1}{2}}}{\frac{1}{2}}+C\).

Step3: Simplify and substitute back

\(\frac{1}{5}\cdot\frac{u^{\frac{1}{2}}}{\frac{1}{2}}+C=\frac{2}{5}\sqrt{u}+C\).
Substituting back \(u = 5x + 8\), we get \(\frac{2}{5}\sqrt{5x+8}+C\).

Answer:

\(\frac{2}{5}\sqrt{5x + 8}+C\)