求定积分dx/(x-a)(x-2a),上限为a/2下限为0

∫(0->π/2) dx/(1+cos²x)
= ∫(0->π/2) dx/(sin²x+cos²x+cos²x)
= ∫(0->π/2) dx/(sin²x+2cos²x)
= ∫(0->π/2) sec²x/(tan²x+2)
= ∫(0->π/2) d(tanx)/(2+tan²x)
= (1/√2)arctan(tanx/√2),(0->π/2)
= (1/√2)arctan[(1/√2)tan(π/2)] - 0
= (1/√2)arctan(∞)
= (1/√2)(π/2)
= π/(2√2)

定积分dx/(e^x+1+e^3-x) 上限正无穷,下限0
=∫(0,+∞)e^x/(e^2x+e^x+e^3)dx
=∫(0,+∞)e^x/((e^x+1/2)^2+e^3-1/4)dx
=1/√(e^3-1/4)*arctan(e^x+1/2)/√(e^3-1/4)|(0,+∞)
=π/2√(e^3-1/4)-1/√(e^3-1/4)*arctan(3/2)/√(e^3-1/4)