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∫ x/(x^2+2x+3) dx
=(1/2)∫ (2x+2)/(x^2+2x+3) dx - ∫ dx/(x^2+2x+3)
=(1/2)ln|x^2+2x+3| -∫ dx/(x^2+2x+3)
=(1/2)ln|x^2+2x+3| -(√2/2)arctan[(x+1)/√2]+C
consider
x^2+2x+3 = (x+1)^2 +2
let
x+1=√2 tanu
dx = √2 (secu)^2 du
∫ dx/(x^2+2x+3)
=(√2/2)∫ du
=(√2/2)u+C'
=(√2/2)arctan[(x+1)/√2]+C'
=(1/2)∫ (2x+2)/(x^2+2x+3) dx - ∫ dx/(x^2+2x+3)
=(1/2)ln|x^2+2x+3| -∫ dx/(x^2+2x+3)
=(1/2)ln|x^2+2x+3| -(√2/2)arctan[(x+1)/√2]+C
consider
x^2+2x+3 = (x+1)^2 +2
let
x+1=√2 tanu
dx = √2 (secu)^2 du
∫ dx/(x^2+2x+3)
=(√2/2)∫ du
=(√2/2)u+C'
=(√2/2)arctan[(x+1)/√2]+C'
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