How do I integrate $$\frac{\sin x+\cos x}{\sin^4 x+\cos^4 x}$$ ? Tried different ways including the tangent half-angle substitution (which seems to be disastrous).
2026-04-13 15:42:33.1776094953
How do I integrate $\frac{\sin x+\cos x}{\sin^4 x+\cos^4 x}$
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\begin{align} \int \frac{\sin x+\cos x}{\sin^4 x + \cos^4 x} \, dx&= \int \frac{\sin x}{\sin^4 x + \cos^4 x} \, dx+\int \frac{\cos x}{\sin^4 x + \cos^4 x} \, dx \\ &= \int \frac{\sin x}{(1-\cos^2 x)^2 + \cos^4 x} \, dx+\int \frac{\cos x}{\sin^4 x + (1-\sin^2 x)^2} \, dx \end{align} Now substitute $u_1=\cos x$ for the first term and $u_2=\sin x$ for the second term.