Question

求解根号下（5-4x-x^2）的不定积分

Answer 1

∫√(5-4x-x^2)dx

=∫√[9-(x+2)^2]dx

x+2=3sinu dx=d(x+2)=d3sinu=3cosudu

=∫9cosu^2du

=(9/2)∫(1+cos2u)du

=(9/2)u+(9/4)sin2u+C

=(9/2)u+(9/2)sinucosu+C

=(9/2)arcsin[(x+2)/3] +(9/2)[(x+2)/3]√(1-(x+2)^2/9) +C

=(9/2)arcsin[(x+2)/3] +(1/2)(x+2)√(5-4x-x^2) +C