int (ln 5 ln 12) dx/(e^x + 4)^(1/2) =
= | t = (e^x + 4)^(1/2); e^x + 4 = t^2; x = ln (t^2 - 4); dx = 2 * t/(t^2 - 4) dt | =
= int (3 4) 2 * t/(t^2 - 4) * 1/t dt = int (3 4) 2/(t^2 - 4) dt =
= int (3 4) 2/((t - 2) * (t + 2)) dt = 1/2 * int (3 4) 4/((t - 2) * (t + 2)) dt =
= 1/2 * int (3 4) ((t + 2) - (t - 2))/((t - 2) * (t + 2)) dt =
= 1/2 * int (3 4) (t + 2) dt/((t - 2) * (t + 2)) - 1/2 * int (3 4) (t - 2) dt/((t - 2) * (t + 2)) =
= 1/2 * int (3 4) dt/(t - 2) - 1/2 * int (3 4) dt/(t + 2) =
= 1/2 * (ln |t - 2|)_{3}^{4} - 1/2 * (ln |t + 2|)_{3}^{4} =
= 1/2 * ln 2 - 1/2 * ln 1 - 1/2 * ln 6 + 1/2 * ln 5 = 1/2 * ln (5/3)