int е^х * ln (1 + 3 * е^х) dx = int ln (1 + 3 * e^x) d(e^x) = | t = e^x | =
= int ln (1 + 3 * t) dt = t * ln (1 + 3 * t) - int t d(ln (1 + 3 * t)) =
= t * ln (1 + 3 * t) - int t * 3/(1 + 3 * t) dt =
= t * ln (1 + 3 * t) - int (3 * t + 1 - 1)/(1 + 3 * t) dt =
= t * ln (1 + 3 * t) - int dt + int dt/(1 + 3 * t) =
= t * ln (1 + 3 * t) - t + 1/3 * int dt/(t + 1/3) =
= t * ln (1 + 3 * t) - t + 1/3 * ln |t + 1/3| + C = | t = e^x | =
= e^x * ln (1 + 3 * e^x) - e^x + 1/3 * ln |e^x + 1/3| + C =
= e^x * ln (1 + 3 * e^x) - e^x + 1/3 * ln (1+ 3 * e^x) + C =
= (e^x + 1/3) * ln (1 + 3 * e^x) - e^x + C