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∫ x/(x² + 2x + 2) dx
= ∫ x/[(x + 1)² + 1] dx
令x + 1 = tany，dx = sec²y dy
= ∫ (tany - 1)/(tan²y + 1) · sec²y dy
= ∫ (tany - 1)/sec²y · sec²y dy
= ∫ (tany - 1) dy
= - ln|cosy| - y + C
= - ln|1/√((x² + 1) + 1)| - arctan(x + 1) + C
= (1/2)ln(x² + 2x + 2) - arctan(x + 1) + C
= ∫ x/[(x + 1)² + 1] dx
令x + 1 = tany，dx = sec²y dy
= ∫ (tany - 1)/(tan²y + 1) · sec²y dy
= ∫ (tany - 1)/sec²y · sec²y dy
= ∫ (tany - 1) dy
= - ln|cosy| - y + C
= - ln|1/√((x² + 1) + 1)| - arctan(x + 1) + C
= (1/2)ln(x² + 2x + 2) - arctan(x + 1) + C
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