Integral of 1/sqrt(5x+10) dx

1. There are multiple ways to do this integral.

   Method #1

   1. Let $u = \sqrt{5 x + 10}$.

      Then let $du = \frac{5 dx}{2 \sqrt{5 x + 10}}$ and substitute $\frac{2 du}{5}$:

      $$\int \frac{2}{5}\, du$$

      1. The integral of a constant times a function is the constant times the integral of the function:

         $$\text{False}$$

         1. The integral of a constant is the constant times the variable of integration:

            $$\int 1\, du = u$$

         So, the result is: $\frac{2 u}{5}$

      Now substitute $u$ back in:

      $$\frac{2 \sqrt{5 x + 10}}{5}$$

   Method #2

   1. Rewrite the integrand:

      $$\frac{1}{\sqrt{5 x + 10}} = \frac{\sqrt{5}}{5 \sqrt{x + 2}}$$

   2. The integral of a constant times a function is the constant times the integral of the function:

      $$\int \frac{\sqrt{5}}{5 \sqrt{x + 2}}\, dx = \frac{\sqrt{5} \int \frac{1}{\sqrt{x + 2}}\, dx}{5}$$

      1. Let $u = x + 2$.

         Then let $du = dx$ and substitute $du$:

         $$\int \frac{1}{\sqrt{u}}\, du$$

         1. The integral of $u^{n}$ is $\frac{u^{n + 1}}{n + 1}$ when $n \neq -1$:

            $$\int \frac{1}{\sqrt{u}}\, du = 2 \sqrt{u}$$

         Now substitute $u$ back in:

         $$2 \sqrt{x + 2}$$

      So, the result is: $\frac{2 \sqrt{5} \sqrt{x + 2}}{5}$

2. Now simplify:

   $$\frac{2 \sqrt{5 x + 10}}{5}$$

3. Add the constant of integration:

   $$\frac{2 \sqrt{5 x + 10}}{5}+ \mathrm{constant}$$

The answer is:

$$\frac{2 \sqrt{5 x + 10}}{5}+ \mathrm{constant}$$