Evaluate. \int \frac{x + 2\sqrt{x - 1}}{2x\sqrt{x - 1}} dx

Question:

Evaluate.

{eq}\displaystyle\; \int \frac{x + 2\sqrt{x - 1}}{2x\sqrt{x - 1}}\,dx {/eq}

Integrals:

We have been given an indefinite integral that has integrand that contains square root term and linear terms. There are many methods of evaluating the integral but sometimes we just need to apply the standard integral formulas.

The given integral:

$$\displaystyle\; \int \frac{x + 2\sqrt{x - 1}}{2x\sqrt{x - 1}}\,dx$$

We will take the constant out of the integation:

$$=\frac{1}{2}\cdot \int \frac{x+2\sqrt{x-1}}{x\sqrt{x-1}}dx\\ =\frac{1}{2}\cdot \int \frac{1}{\sqrt{x-1}}+\frac{2}{x}dx\\ =\frac{1}{2}\left(\int \frac{1}{\sqrt{x-1}}dx+\int \frac{2}{x}dx\right)$$

We will apply the standard integral formulas:

$$=\frac{1}{2}\left(2\sqrt{x-1}+2\ln \left|x\right|\right)\\ =\sqrt{x-1}+\ln \left|x\right|+C$$

C is the constant of integration.