Question

$\int \frac{\sin 2x}{\sqrt{{\sin }^{4}x+4{\sin }^{2}x-2}}$dx

Answer 1

$\int \frac{sin2x}{si{n}^{4}x+4si{n}^{2}x-2}dx$

Let $si{n}^{2}x=t$

$2sinxcosx=dt/dx$

$sin2x=dt/dx$

$sin2xdx=dt$

$\int \frac{dt}{\sqrt{(t{)}^2+4t-2}}$ $\left[\right(si{n}^{4}x)=(si{n}^{2}x{)}^2]$

Now completing whole square in denominator

$\int \frac{dt}{\sqrt{t^2+4t-2+4-4}}$

$\int \frac{dt}{t^2-4t+4-4-2}$

$\int \frac{dt}{\sqrt{t^2-2t-2t+4-6}}=\int \frac{dt}{\sqrt{t(t-2)-2(t-2)-6}}$

$=\int \frac{dt}{\sqrt{(t-2{)}^2-6}}$

$=\int \frac{dt}{\sqrt{(t-2{)}^2-(\sqrt{6}{)}^2}}$ $[\frac{1}{\sqrt{x^2-a^2}}dx=log\left|\sqrt{x^2-a^2+x}\right|+c]$

$=log\left|\sqrt{(t-2{)}^2-(\sqrt{6}{)}^2+(t-2)}\right|+c$

$t=si{n}^{2}x$

$=log\left|\sqrt{(si{n}^{2}x-2{)}^2-(\sqrt{6}{)}^2+(si{n}^{2}x-2)}\right|+c$

$=log\left|\sqrt{si{n}^{4}x-4si{n}^{2}x-2+(si{n}^{2}x-2)}\right|+c$