Необходимо найти такой интеграл:
int (5 * ln x - (ln x)^3 + 1)/(x * (5 - (ln x)^2)) dx.
Я пыталась заменить lnx на t и подставить эту переменную, но что то не клеится.
int (5 * ln x - (ln x)^3 + 1)/(x * (5 - (ln x)^2)) dx = | ln x = t; x = e^t; dx = e^t dt | =
= int (5 * t - t^3 + 1)/(e^t * (5 - t^2)) * e^t dt = int (5 * t - t^3 + 1)/(5 - t^2) dt =
= int (5 * t - t^3)/(5 - t^2) dt + int 1/(5 - t^2) dt =
= int t * (5 - t^2)/(5 - t^2) dt + int 1/(5 - t^2) dt =
= int t dt + int 1/(5 - t^2) dt = 1/2 * t^2 + int 1/((5^(1/2) - t) * (5^(1/2) + t)) dt =
= 1/2 * t^2 + 1/(2 * 5^(1/2)) * int 2 * 5^(1/2)/((5^(1/2) - t) * (5^(1/2) + t)) dt =
= 1/2 * t^2 + 1/(2 * 5^(1/2)) *
* int ((5^(1/2) - t) + (5^(1/2) + t))/((5^(1/2) - t) * (5^(1/2) + t)) dt =
= 1/2 * t^2 + 1/(2 * 5^(1/2)) * int 1/(t + 5^(1/2)) dt -
- 1/(2 * 5^(1/2)) * int 1/(t - 5^(1/2)) dt =
= 1/2 *t^2 + 1/(2 * 5^(1/2)) * ln |t + 5^(1/2)| - 1/(2 * 5^(1/2)) * ln |t - 5^(1/2)| + C =
= 1/2 * t^2 + 1/(2 * 5^(1/2)) * ln (|t + 5^(1/2)|/|t - 5^(1/2)|) + C =
= | t = ln x | = 1/2 * ln^2 x + 1/(2 * 5^(1/2)) * ln (|ln x + 5^(1/2)|/|ln x - 5^(1/2)|) + C
Спасибо большое.
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