Question

$\int\ { \frac{1}{ sinx^{4}+sin x^{4}}\,dx$

d1f4a0649434d38decb730fad0866dd9.docx

Answer 1

$\int {\frac{1}{Sin^4\ x\ +\ Cos^4\ x} \, dx = \int {\frac{1}{(Sin^2\ x\ +\ Cos^2\ x)^2 - 2 sin^2\ x\ cos^2 x} \, dx$

$\int {\frac{1} {1 - \frac{1}{2}Sin^2 2x} } \, dx \\ \\ \int {\frac{2}{2-sin^2\ 2x}} \, dx = \int {\frac{2}{1+cos^2\ 2x}} \, dx = \int {\frac{2*2}{3 + cos\ 4x}} \, dx$

$let\ tan\ 2x\ =\ y, \ \ \ Sec^2\ 2x\ dx = dy,\ \ \ dx = \frac{dy}{1+y^2} \\ \\ 3+ Cos\ 4x = 3 + \frac{1-y^2}{1+y^2} = \frac{2(2 + y^2)}{1+y^2} \\ \\ Integral = \int {\frac{4(1+y^2)}{2(2+y^2)}} \, \frac{dt}{1+y^2} \\ \\ = \int {\frac{2}{2+y^2}} \, dy = \sqrt2*tan^{-1}\ (y/\sqrt2) \\ \\ \sqrt2 * tan^{-1} (\frac{1}{\sqrt2}*tan\ 2x)$