为自己加油

计算下列定积分
(1)∫_-1^√3[1/(1+x²)]dx
(2)∫₀¹[3/√(3x+1)]dx
(3)∫_-(π/2)^π/2(x³+3)cos2xdx
(4)∫₀¹[x³/√(4-x²)]dx
(5)∫₀^π/2√(1-sin2x)dx
(6)∫_π/4^π/2(x/sin²x)dx

(1)∫_-1^√31/(1+x²)dx =arctanx|_-1^√3=π/3-(-π/4)=7π/12 (2)∫₀¹3dx/√(3x+1) =2∫₀¹d(3x+1)/[2√(3x+1)] =2√(3x+1)|₀¹=2×(2-1)=2 (3)由于积分区间为对称区间，x³cos2x为奇函数，cos2x为偶函数，故 ∫_-(π/2)^π/2(x²+3)cos2xdx =∫_-(π/2)^π/2x³cos2xdx+∫_-(π/2)^π/23cos2xdx =0+6∫₀^π/2cos2xdx-3sin2x|₀^π/2=0 (4)令x=2sint，则dx=2costdt ∫₀¹x³/[√(4-x²)]dx =∫₀^π/68sin³t/√(4-4sin²t)•2costdt =∫₀^π/68sin³tdt =-8∫₀^π/6(1-cos²t)d(cost) =-8(cost-1/3cos³t)|₀^π/6=16/3-3√3 (5)∫₀^π/2√1-sin2xdx=∫₀^π/2√(sinx-cosx)²dx =∫₀^π/2|cosx-sinx|dx =∫₀^π/4(cosx-sinx)dx+∫_π/4^π/2sinx-cosx)dx =(sinx+cosx)|₀^π/4-(cosx+sinx)|_π/4^π/2 =2√2-2 (6)∫_π/4^π/2(x/sin²x)dx=-∫_π/4^π/2xdcotx=-(xcotx)|_π/4^π/2+∫_π/4^π/2cotxdx =π/4+lnsinx|_π/4^π/2=π/4+1/2ln2