Question

Integrate:

$\int \frac{\tan x}{a+b{\tan }^{2}x}dx$

Answer 1

$\int \frac{tanx}{a+bta{n}^{2}x}dx$

$\Rightarrow \int \frac{\frac{sinx}{cosx}}{a+b\frac{si{n}^{2}x}{co{s}^{2}x}}dx=\int \frac{sinx}{\frac{aco{s}^{2}x+bsi{n}^{2}x}{cosx}}dx$

$\Rightarrow \int \frac{sinx.cosx}{a.a^{2}si{n}^{2}x+bsi{n}^{2}x}dx$

$\Rightarrow \int \frac{sinx.cosx}{a+(b-a)si{n}^{2}x}dx$

put $a+(b-a)si{n}^{2}x=t$

$2(b-a)sinx.cosxdx=dt$

so,$sinx.cosxdx=\frac{dt}{2(b-a)}$

$\Rightarrow \frac{1}{2(b-a)}\int \frac{1}{t}dt$

$\Rightarrow \frac{1}{2(b-a)}lnt+c$

$\Rightarrow \frac{1}{2(b-a)}ln[a+(b-a\left)si{n}^{2}n\right]+c$