Question

Solve:$\int \frac{si{n}^{8}x-co{s}^{8}x}{1-2si{n}^{2}xco{s}^{2}x}dx$

Answer 1

$\int \frac{si{n}^{8}x-co{s}^{8}x}{1-2si{n}^{2}co{s}^{2}x}dx$

$=\int \frac{(si{n}^{4}x-co{s}^{4}x)(si{n}^{4}x+co{s}^{4}x)}{(si{n}^{2}x+co{s}^{2}x{)}^2-2si{n}^{2}xco{s}^{2}x}dx$

$=\int \frac{(si{n}^{2}x+co{s}^{2}x)(si{n}^{2}x-co{s}^{2}x)(si{n}^{4}x+co{s}^{4}x)}{si{n}^{4}x+co{s}^{4}x+2si{n}^{2}xco{s}^{2}x-2si{n}^{2}xco{s}^{2}x}dx$

$=-\int 1(co{s}^{2}x-si{n}^{2}x)dx$

$=-\int cos2xdx=-\left[\frac{sin2x}{2}\right]+c$

$=\frac{-sin2x}{2}+c$

where c is the constant of integration