Добрый день! Наведите на мысли по решению следующих интегралов
1. int dx/(x^2 + x + 1)
2. int x^3 * ln x dx
Помогите, плз.
1. int dx/(x^2 + x + 1) = int dx/(x^2 + 2 * 1/2 * x * 1/4 + 3/4) =
= int dx/((x + 1/2)^2 + 3/4) = int d(x + 1/2)/((x + 1/2)^2 + 3/4) =
= | t = x + 1/2 | = int dt/(t^2 + 3/4) = 2/3^(1/2) * arctg (2 * t/3^(1/2)) + C =
= | t = x + 1/2 | = 2/3^(1/2) * arctg ((2x + 1)/3^(1/2)) + C
2. int x^3 * ln x dx = int ln x d(1/4 * x^4) = 1/4 * int ln x d(x^4) =
= 1/4 * x^4 * ln x - 1/4 * int x^4 d(ln x) =
= 1/4 * x^4 * ln x - 1/4 * int x^3 dx = 1/4 * x^4 * ln x - 1/4 * 1/4 * x^4 + C =
= 1/4 * x^4 * ln x - 1/16 * x^4 + C
Спасибо Вам большое.
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