Evaluate ∫1x4+1 dx\int \frac{1}{x^4+1} \,dx∫x4+11dx. (Hint: use partial fractions or algebraic manipulation)
142ln∣x2+2x+1x2−2x+1∣+122arctan(2x+1)+122arctan(2x−1)+C\frac{1}{4\sqrt{2}} \ln|\frac{x^2+\sqrt{2}x+1}{x^2-\sqrt{2}x+1}| + \frac{1}{2\sqrt{2}} \arctan(\sqrt{2}x+1) + \frac{1}{2\sqrt{2}} \arctan(\sqrt{2}x-1) + C421ln∣x2−2x+1x2+2x+1∣+221arctan(2x+1)+221arctan(2x−1)+C
142ln∣x2−2x+1x2+2x+1∣+122(arctan(2x+1)+arctan(2x−1))+C\frac{1}{4\sqrt{2}} \ln|\frac{x^2-\sqrt{2}x+1}{x^2+\sqrt{2}x+1}| + \frac{1}{2\sqrt{2}} (\arctan(\sqrt{2}x+1) + \arctan(\sqrt{2}x-1)) + C421ln∣x2+2x+1x2−2x+1∣+221(arctan(2x+1)+arctan(2x−1))+C
122ln∣x2+2x+1x2−2x+1∣+C\frac{1}{2\sqrt{2}} \ln|\frac{x^2+\sqrt{2}x+1}{x^2-\sqrt{2}x+1}| + C221ln∣x2−2x+1x2+2x+1∣+C