Question

If the integral $I=\int \frac{tanx}{5+7ta{n}^{2}x}dx$ $=kln\left|f\left(x\right)\right|+C$ (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

Answer: 6

$I=\int \frac{sinxcosx}{5co{s}^{2}x+7si{n}^{2}x}dx$
Let $si{n}^{2}x=t$
$\Rightarrow sinxcosxdx=\frac{dt}{2}$
$I=\frac{1}{2}\int \frac{dt}{5\left(1-t\right)+7t}$
$=\frac{1}{2}\int \frac{dt}{5+2t}$
$=\frac{1}{4}ln\left|5+2t\right|+C$
$=\frac{1}{4}ln\left|5+2si{n}^{2}x\right|+C$
$\therefore f\left(x\right)=5+2{\left(sin\right)}^{2}x$
$\Rightarrow f\left(\frac{\pi }{4}\right)=6$