Question

If the integral $I=\displaystyle \int \frac{tan x}{5 + 7 tan^{2} ⁡ x}dx$ $=kln \left|f \left(x\right)\right|+C$ (where $C$ is the integration constant) and $f\left(0\right)=5$ , then the value of $f\left(\frac{\pi }{4}\right)$ is equal to

Answer 1

$I=\displaystyle \int \frac{sin x cos ⁡ x}{5 cos^{2} ⁡ x + 7 sin^{2} ⁡ x} d x$
Let $sin^{2} x=t$
$\Rightarrow sin xcos ⁡ xdx=\frac{d t}{2}$
$I=\frac{1}{2}\displaystyle \int \frac{d t}{5 \left(1 - t\right) + 7 t}$
$=\frac{1}{2}\displaystyle \int \frac{d t}{5 + 2 t}$
$=\frac{1}{4}ln \left|5 + 2 t\right| + C$
$=\frac{1}{4}ln \left|5 + 2 sin^{2} ⁡ x\right| + C$
$\therefore f\left(x\right)=5+2\left(sin\right)^{2} x$
$\Rightarrow f\left(\frac{\pi }{4}\right)=6$