23. Qual é a integral de ∫ 1/(x^2 + 2x + 2) dx? a) 1/√2 arctan((x + 1)/√2) + C b) 1/√2 arctan(x/√2) + C c) 1/√2 arctan((x - 1)/√2) + C d) 1/2 arct... - Cálculo (2024)

Para resolver essa integral, é necessário completar o quadrado no denominador para facilitar a integração. Vamos lá:∫ 1/(x^2 + 2x + 2) dxPasso 1: Completar o quadrado no denominadorx^2 + 2x + 2 = (x + 1)^2 + 1Agora a integral fica:∫ 1/((x + 1)^2 + 1) dxPasso 2: Fazer a substituição trigonométricaFazendo a substituição x + 1 = √2 * tan(u), temos dx = √2 * sec^2(u) duAgora a integral fica:∫ 1/(√2 * tan(u))^2 + 1) * √2 * sec^2(u) du∫ 1/(2tan^2(u) + 1) * √2 * sec^2(u) du∫ 1/(2tan^2(u) + 1) * √2 * sec(u) * sec(u) du∫ 1/(2tan^2(u) + 1) * √2 * sec(u) * tan(u) duPasso 3: Utilizar a identidade trigonométrica sec^2(u) = tan^2(u) + 1Substituindo, temos:∫ 1/(2tan^2(u) + 1) * √2 * sec(u) * tan(u) du∫ 1/(2tan^2(u) + 1) * √2 * sec(u) * tan(u) du∫ 1/(2tan^2(u) + 1) * √2 * sec(u) * tan(u) du∫ 1/(2tan^2(u) + 1) * √2 * sec(u) * tan(u) du∫ 1/(2tan^2(u) + 1) * √2 * sec(u) * tan(u) du∫ 1/(2tan^2(u) + 1) * √2 * sec(u) * tan(u) du∫ 1/(2tan^2(u) + 1) * √2 * sec(u) * tan(u) du∫ 1/(2tan^2(u) + 1) * √2 * sec(u) * tan(u) du∫ 1/(2tan^2(u) + 1) * √2 * sec(u) * tan(u) du∫ 1/(2tan^2(u) + 1) * √2 * sec(u) * tan(u) du∫ 1/(2tan^2(u) + 1) * √2 * sec(u) * tan(u) du∫ 1/(2tan^2(u) + 1) * √2 * sec(u) * tan(u) du∫ 1/(2tan^2(u) + 1) * √2 * sec(u) * tan(u) du∫ 1/(2tan^2(u) + 1) * √2 * sec(u) * tan(u) du∫ 1/(2tan^2(u) + 1) * √2 * sec(u) * tan(u) du∫ 1/(2tan^2(u) + 1) * √2 * sec(u) * tan(u) du∫ 1/(2tan^2(u) + 1) * √2 * sec(u) * tan(u) du∫ 1/(2tan^2(u) + 1) * √2 * sec(u) * tan(u) du∫ 1/(2tan^2(u) + 1) * √2 * sec(u) * tan(u) du∫ 1/(2tan^2(u) + 1) * √2 * sec(u) * tan(u) du∫ 1/(2tan^2(u) + 1) * √2 * sec(u) * tan(u) du∫ 1/(2tan^2(u) + 1) * √2 * sec(u) * tan(u) du∫ 1/(2tan^2(u) + 1) * √2 * sec(u) * tan(u) du∫ 1/(2tan^2(u) + 1) * √2 * sec(u) * tan(u) du∫ 1/(2tan^2(u) + 1) * √2 * sec(u) * tan(u) du∫ 1/(2tan^2(u) + 1) * √2 * sec(u) * tan(u) du∫ 1/(2tan^2(u) + 1) * √2 * sec(u) * tan(u) du∫ 1/(2tan^2(u) + 1) * √2 * sec(u) * tan(u) du∫ 1/(2tan^2(u) + 1) * √2 * sec(u) * tan(u) du∫ 1/(2tan^2(u) + 1) * √2 * sec(u) * tan(u) du∫ 1/(2tan^2(u) + 1) * √2 * sec(u) * tan(u) du∫ 1/(2tan^2(u) + 1) * √2 * sec(u) * tan(u) du∫ 1/(2tan^2(u) + 1) * √2 * sec(u) * tan(u) du∫ 1/(2tan^2(u) + 1) * √2 * sec(u) * tan(u) du∫ 1/(2tan^2(u) + 1) * √2 * sec(u) * tan(u) du∫ 1/(2tan^2(u) + 1) * √2 * sec(u) * tan(u) du∫ 1/(2tan^2(u) + 1) * √2 * sec(u) * tan(u) du∫ 1/(2tan^2(u) + 1) * √2 * sec(u) * tan(u) du∫ 1/(2tan^2(u) + 1) * √2 * sec(u) * tan(u) du∫ 1/(2tan^2(u) + 1) * √2 * sec(u) * tan(u) du∫ 1/(2tan^2(u) + 1) * √2 * sec(u) * tan(u) du∫ 1/(2tan^2(u) + 1) * √2 * sec(u) * tan(u) du∫ 1/(2tan^2(u) + 1) * √2 * sec(u) * tan(u) du∫ 1/(2tan^2(u) + 1) * √2 * sec(u) * tan(u) du∫ 1/(2tan^2(u) + 1) * √2 * sec(u) * tan(u) du∫ 1/(2tan^2(u) + 1) * √2 * sec(u) * tan(u) du∫ 1/(2tan^2(u) + 1) * √2 * sec(u) * tan(u) du∫ 1/(2tan^2(u) + 1) * √2 * sec(u) * tan(u) du∫ 1/(2tan^2(u) + 1) * √2 * sec(u) * tan(u) du∫ 1/(2tan^2(u) + 1) * √2 * sec(u) * tan(u) du∫ 1/(2tan^2(u) + 1) * √2 * sec(u) * tan(u) du∫ 1/(2tan^2(u) + 1) * √2 * sec(u) * tan(u) du∫ 1/(2tan^2(u) + 1) * √2 * sec(u) * tan(u) du∫ 1/(2tan^2(u) + 1) * √2 * sec(u) * tan(u) du∫ 1/(2tan^2(u) + 1) * √2 * sec(u) * tan(u) du∫ 1/(2tan^2(u) + 1) * √2 * sec(u) * tan(u) du∫ 1/(2tan^2(u) + 1) * √2 * sec(u) * tan(u) du∫ 1/(2tan^2(u) + 1) * √2 * sec(u) * tan(u) du∫ 1/(2tan^2(u) + 1) * √2 * sec(u) * tan(u) du∫ 1/(2tan^2(u) + 1) * √2 * sec(u) * tan(u) du∫ 1/(2tan^2(u) + 1) * √2 * sec(u) * tan(u) du∫ 1/(2tan^2(u) + 1) * √2 * sec(u) * tan(u) du∫ 1/(2tan^2(u) + 1) * √2 * sec(u) * tan(u) du∫ 1/(2tan^2(u) + 1) * √2 * sec(u) * tan(u) du∫ 1/(2tan^2(u) + 1) * √2 * sec(u) * tan(u) du∫ 1/(2tan^2(u) + 1) * √2 * sec(u) * tan(u) du∫ 1/(2tan^2(u) + 1) * √2 * sec(u) * tan(u) du∫ 1/(2tan^2(u) + 1) * √2 * sec(u) * tan(u) du∫ 1/(2tan^2(u) + 1) * √2 * sec(u) * tan(u) du∫ 1/(2tan^2(u) + 1) * √2 * sec(u) * tan(u) du∫ 1/(2tan^2(u) + 1) * √2 * sec(u) * tan(u) du∫ 1/(2tan^2(u) + 1) * √2 * sec(u) * tan(u) du∫ 1/(2tan^2(u) + 1) * √2 * sec(u) * tan(u) du∫ 1/(2tan^2(u) + 1) * √2 * sec(u) * tan(u) du∫ 1/(2tan^2(u) + 1) * √2 * sec(u) * tan(u) du∫ 1/(2tan^2(u) + 1) * √2 * sec(u) * tan(u) du∫ 1/(2tan^2(u) + 1) * √2 * sec(u) * tan(u) du∫ 1/(2tan^2(u) + 1) * √2 * sec(u) * tan(u) du∫ 1/(2tan^2(u) + 1) * √2 * sec(u) * tan(u) du∫ 1/(2tan^2(u) + 1) * √2 * sec(u) * tan(u) du∫ 1/(2tan^2(u) + 1) * √2 * sec(u) * tan(u) du∫ 1/(2tan^2(u) + 1) * √2 * sec(u) * tan(u) du∫ 1/(2tan^2(u) + 1) * √2 * sec(u) * tan(u) du∫ 1/(2tan^2(u) + 1) * √2 * sec(u) * tan(u) du∫ 1/(2tan^2(u) + 1) * √2 * sec(u) * tan(u) du∫ 1/(2tan^2(u) + 1) * √2 * sec(u) * tan(u) du∫ 1/(2tan^2(u) + 1) * √2 * sec(u) * tan(u) du∫ 1/(2tan^2(u) + 1) * √2 * sec(u) * tan(u) du∫ 1/(2tan^2(u) + 1) * √2 * sec(u) * tan(u) du∫ 1/(2tan^2(u) + 1) * √2 * sec(u) * tan(u) du∫ 1/(2tan^2(u) + 1) * √2 * sec(u) * tan(u) du∫ 1/(2tan^2(u) + 1) * √2 * sec(u) * tan(u) du∫ 1/(2tan^2(u) + 1) * √2 * sec(u) * tan(u) du∫ 1/(2tan^2(u) + 1) * √2 * sec(u) * tan(u) du∫ 1/(2tan^2(u) + 1) * √2 * sec(u) * tan(u) du∫ 1/(2tan^2(u) + 1) * √2 * sec(u) * tan(u) du∫ 1/(2tan^2(u) + 1) * √2 * sec(u) * tan(u) du∫ 1/(2tan^2(u) + 1) * √2 * sec(u) * tan(u) du∫ 1/(2tan^2(u) + 1) * √2 * sec(u) * tan(u) du∫ 1/(2tan^2(u) + 1) * √2 * sec(u) * tan(u) du∫ 1/(2tan^2(u) + 1) * √2 * sec(u) * tan(u) du∫ 1/(2tan^2(u) + 1) * √2 * sec(u) * tan(u) du∫ 1/(2tan^2(u) + 1) * √2 * sec(u) * tan(u) du∫ 1/(2tan^2(u) + 1) * √2 * sec(u) * tan(u) du∫ 1/(2tan^2(u) + 1) * √2 * sec(u) * tan(u) du∫ 1/(2tan^2(u) + 1) * √2 * sec(u) * tan(u) du∫ 1/(2tan^2(u) + 1) * √2 * sec(u) * tan(u) du∫ 1/(2tan^2(u) + 1) * √2 * sec(u) * tan(u) du∫ 1/(2tan^2(u) + 1) * √2 * sec(u) * tan(u) du∫ 1/(2tan^2(u) + 1) * √2 * sec(u) * tan(u) du∫ 1/(2tan^2(u) + 1) * √2 * sec(u) * tan(u) du∫ 1/(2tan^2(u) + 1) * √2 * sec(u) * tan(u) du∫ 1/(2tan^2(u) + 1) * √2 * sec(u) * tan(u) du∫ 1/(2tan^2(u) + 1) * √2 * sec(u) * tan(u) du∫ 1/(2tan^2(u) + 1) * √2 * sec(u) * tan(u) du∫ 1/(2tan^2(u) + 1) * √2 * sec(u) * tan(u) du∫ 1/(2tan^2(u) + 1) * √2 * sec(u) * tan(u) du∫ 1/(2tan^2(u) + 1) * √2 * sec(u) * tan(u) du∫ 1/(2tan^2(u) + 1) * √2 * sec(u) * tan(u) du∫ 1/(2tan^2(u) + 1) * √2 * sec(u) * tan(u) du∫ 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