Evaluate ∫0111+x4dx\int_0^1 \frac{1}{1+x^4} dx∫011+x41dx using the partial fraction decomposition for x4+1=(x2+2x+1)(x2−2x+1)x^4+1 = (x^2+\sqrt{2}x+1)(x^2-\sqrt{2}x+1)x4+1=(x2+2x+1)(x2−2x+1).
142ln(1+2)+122arctan(1+2)\frac{1}{4\sqrt{2}} \ln(1+\sqrt{2}) + \frac{1}{2\sqrt{2}} \arctan(1+\sqrt{2})421ln(1+2)+221arctan(1+2)
122ln(1+2)+122arctan(1)\frac{1}{2\sqrt{2}} \ln(1+\sqrt{2}) + \frac{1}{2\sqrt{2}} \arctan(1)221ln(1+2)+221arctan(1)
142ln(3+22)+122arctan(1)\frac{1}{4\sqrt{2}} \ln(3+2\sqrt{2}) + \frac{1}{2\sqrt{2}} \arctan(1)421ln(3+22)+221arctan(1)
π4\frac{\pi}{4}4π