Решите определенный интеграл
[tex] \int\limits { \frac{dx}{ \sqrt{(5+2x+x^2)^3} } } \ [/tex]
[tex] \int\limits { \frac{dx}{ \sqrt{(5+2x+x^2)^3} } } \ [/tex]
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X^2 + 2x + 5 = x^2 + 2x + 1 + 4 = (x + 1)^2 + 4Замена x+1 = 2*tg y, x = 2tg y - 1, dx = -2/cos^2 y dy = -2(1 + tg^2 y) dy
√(x^2+2x+5)^3 = (4tg^2 y + 4)^(3/2) = 4^(3/2)*(1 + tg^2 y)^(3/2) = 8(1 + tg^2 y)^(3/2)
Int dx / √(x^2+2x+5)^3 = -2*Int (1 + tg^2 y) dy / (8(1 + tg^2 y)^(3/2)) =
-1/4*Int dy / √(1 + tg^2 y) = -1/4*Int dy* √(cos^2 y) = -1/4* Int cos y dy =
= -1/4*sin y + C = -1/4*sin(arctg (x+1)/2) + C