Question

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

Answer 1

Consider the given integral.

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

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

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

$I=\int ({\sin }^{2}x-{\cos }^{2}x)\times 1dx$

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

$I=-\int \cos 2xdx$

$I=-\left[\frac{\sin 2x}{2}\right]+C$