∫(-1到1) dx/(x²+1)²
= 2∫(0到1) dx/(x²+1)²
令x=tanz,dx=sec²z dz
当x=0,z=0 // 当x=1,z=π/4
= 2∫(0到π/4) (sec²z)/(sec⁴z) dz
= 2∫(0到π/4) cos²z dz
= ∫(0到π/4) (1+cos2z) dz
= (z+1/2*sin2z)[0到π/4]
= [π/4+1/2]
= (2+π)/4
∫(-1到1) dx/(x²+1)²
= 2∫(0到1) dx/(x²+1)²
令x=tanz,dx=sec²z dz
当x=0,z=0 // 当x=1,z=π/4
= 2∫(0到π/4) (sec²z)/(sec⁴z) dz
= 2∫(0到π/4) cos²z dz
= ∫(0到π/4) (1+cos2z) dz
= (z+1/2*sin2z)[0到π/4]
= [π/4+1/2]
= (2+π)/4