帮忙算下积分.∫dx/(x+√(a^2-x^2)).

Question

帮忙算下积分.∫dx/(x+√(a^2-x^2)).

Answer 1

∫dx/(x+√a^2-x^2)
=∫d(x/a)/(x/a+√(1-(x/a)^2) )
x/a=sinu
=∫dsinu/(cosu+sinu)
=(1/√2)∫cosudu/sin(u+π/4)
=(1/√2)∫[cos(u+π/4)cos(π/4)+sin(u+π/4)sin(π/4)]du/sin(u+π/4)
=(1/2)∫cos(u+π/4)du/sin(u+π/4)+ (1/2)u
=(1/2)ln|sin(u+π/4)|+(1/2)u +C
=(1/2)ln|√2[x/a+√(1-x^2/a^2)]|+(1/2)u+C0
=(1/2)ln|x+√(a^2-x^2)|+(1/2)arcsin(x/a)+C