The usage of derivation formulas.

We use here Tables 4.4 and 1.5.

Example 8.2.1   Compute $F(x)= \int \frac {2x+3}{2\sqrt{x^2+3x+5}} \; dx$ .

Define $u(x)=x^2+3x+5$ ; then $u'(x)=2x+3$ and the integrand is of the form $\frac {u'(x)}{2 \sqrt{u(x)}}$ . Therefore $F(x)=\sqrt{x^2+3x+5} +C, \; C \in \mathbb{R}$ .

Example 8.2.2   Compute $F(x)= \int \frac {2x+1}{x^2+x+2} \; dx$ .

Define $u(x)=x^2+x+2$ ; then $u'(x)=2x+1$ and the integrand is of the form $\frac {u'(x)}{u(x)}$ . Note that the quadratic polynomial $x^2+x+2$ does not vanish on $\mathbb{R}$ .

Therefore $F(x)=\ln (x^2+x+2) +C, \; C \in \mathbb{R}$ .

Example 8.2.3   Compute $F(x)= \int (\tan x + \tan^3 x) \; dx$ , for $-\pi /2 < x < \pi /2$ .

We have: $F(x)= \int \tan x (1+ \tan^2 x) \; dx$ . Define $u(x)=\tan x$ ; $u'(x)=1+ \tan^2 x$ and the integrand is of the form $u' \cdot u = \frac 12 (u^2)'$ .

Therefore $\forall x \in \left( -\frac {\pi}{2}, \frac {\pi}{2} \right), \;F(x)= \tan x +C, \; C \in \mathbb{R}$ .