Question

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

Answer 1

We have,

$I=\int \frac{{\sin }^{8}x-{\cos }^{8}x}{1-2{\sin }^{2}x{\cos }^{2}x}dx$

Then,

$I=\int \frac{{\left({\sin }^{4}x\right)}^2-{\left({\cos }^{4}x\right)}^2}{{({\sin }^{2}x+{\cos }^{2}x)}^2-2{\sin }^{2}x{\cos }^{2}x}dx$

$I=\int \frac{({\sin }^{4}x-{\cos }^{4}x)({\sin }^{4}x+{\cos }^{4}x)}{({\sin }^{4}x+{\cos }^{4}x)}dx$

$I=\int ({\sin }^{4}x-{\cos }^{4}x)dx$

$I=\int ({\sin }^{2}x-{\cos }^{2}x)({\sin }^{2}x+{\cos }^{2}x)dx$

$I=\int ({\sin }^{2}x-{\cos }^{2}x)dx$

$I=-\int \cos 2xdx$

$I=-\frac{\sin 2x}{2}+C$

Hence, this is the answer.