为自己加油

计算下列定积分：
(1)∫_-1√3^√3[1/(1+x²)]dx
(2)∫₀¹dx/√(4-x²)；
(3)∫₀¹α^xe^xdx
(4)∫₀¹[x⁴/(1+x²)]dx
(5)∫_-(π/2)^π/2sin²(x/2)dx；
(6)∫₀^π/4tan²xdx；
(7)∫₀^2π|sinx|dx；
(8)∫₀³|2-x|dx
(9)∫_-1²f(x)dx，其中f(x)=
{x，x∈[0,2]
{x²，x∈[-1,0]
(10)∫₀^πsin³xcos²xdx；
(11)∫₁^e[(1+lnx)/x]dx
(12)∫₁²[dx/(x+x²)]；
(13)∫₀¹(arcsin√x)/√[x(1-x)]dx

(1)∫_-1/√3^√3[1/(1+x²)]dx=arctanx|_-1/√3^√3=π/2 (2)∫₀¹dx/√(4-x²) =∫₀¹d(x/2)/√[1-(x/2)²] =arcsin(x/2)|₀¹=π/6． (3)∫₀¹α^xe^xdx=∫₀¹(αe)^xdx=[(αe)^x/lnαe]|₀¹=(αe-1)/[ln(αe)] (4)∫₀¹[x⁴/(1+x²)]dx=∫₀¹(x²-1)dx+∫₀¹[1/(1+x²)]dx =[x³/2-x]|₀¹+arctanx|₀¹=π/4-2/3 (5)∫_-(π/2)^π/2sin²(x/2)dx=∫_-(π/2)^π/2[(1-cosx)/2]dx=[(x-sinx)/2]|_-(π/2)^π/2=π/2-1 (6)∫₀^π/4tan²xdx=∫₀^π/4(sec²x-1)dx=[tanx-x]|₀^π/4=1-π/4 (7)∫₀^2π|sinx|dx=∫₀^πsinxdx-∫_π^2πsinxdx =-cosx|₀^π/4+cosx|_π^2π=4 (8)∫₀³|2-x|dx=∫₀²(2-x)dx+∫₂³(x-2)dx= [2x-(1/2)x²]|₀²+[x²/2-2x]|₂³=5/2 (9)∫_-1²f(x)dx=∫_-1⁰x²dx+∫₀²xdx=(1/3)x³|_-1⁰+(1/2)x²|₀²=7/3 (10)₀^πsin³xcos²xdx=1/4∫₀^πcosxdsin4x =(1/4)cosxsin4x|₀^π-1/4∫₀^π(1-cos²x)²dcosx =-1/4[cosx-2/3 cos²x+1/5 cos⁵x]|_π⁰ =4/15 (11)∫₁^e[(1+lnx)/x]dx=∫₁^e(1+lnx)dlnx=[lnx+(1/2)ln²x]|₁^e=3/2 (12)∫₂¹[dx/(x+x²)]=∫₁²[1/x-1/(x+1)]dx=ln|x/(x+1)||₁²=ln(4/3) (13)∫₀¹[arcsin√x/√[x(1-x)] =2∫₀¹arcsin√xdarcsin√x =arcsin²√x|₀¹ =(π/4)²=π²/4