Examples

Example 2.2.3  

$$\int x^2 + 1 + \frac{1}{x^2 + 1} dx = \int x^2dx + \int 1dx + \int \frac{1}{x^2+1} dx$$

$$= \frac{1}{3} x^2 + x + \tan^{-1}(x) + c.$$

Example 2.2.4  

$$\int \sqrt{\frac{5}{x}} dx = \int \sqrt{5} x^{-1/2} dx = 2\sqrt{5} x^{1/2} + c.$$

Example 2.2.5  

$$\int \frac{\sin(2x)}{\sin(x)} dx = \int \frac{2\sin(x)\cos(x)}{\sin(x)} = \int 2\cos(x) = 2\sin(x) + c$$