求∫(2x+3)/(x²+2x+2)dx
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2x+3= (2x+2) +1
∫(2x+3)/(x^2+2x+2)dx
=∫(2x+2)/(x^2+2x+2)dx +∫dx/(x^2+2x+2)
=ln|x^2+2x+2| +∫dx/(x^2+2x+2)
consider
x^2+2x+2 =(x+1)^2 +1
let
x+1= tany
dx =(secy)^2. dy
∫dx/(x^2+2x+2)
=∫ dy
=y
= arctan(x+1)
∫(2x+3)/(x^2+2x+2)dx
=ln|x^2+2x+2| +∫dx/(x^2+2x+2)
=ln|x^2+2x+2| +arctan(x+1) + C
∫(2x+3)/(x^2+2x+2)dx
=∫(2x+2)/(x^2+2x+2)dx +∫dx/(x^2+2x+2)
=ln|x^2+2x+2| +∫dx/(x^2+2x+2)
consider
x^2+2x+2 =(x+1)^2 +1
let
x+1= tany
dx =(secy)^2. dy
∫dx/(x^2+2x+2)
=∫ dy
=y
= arctan(x+1)
∫(2x+3)/(x^2+2x+2)dx
=ln|x^2+2x+2| +∫dx/(x^2+2x+2)
=ln|x^2+2x+2| +arctan(x+1) + C
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