Вычислить определенный интеграл.

amanda2sempl: ∫ln(x+1)dx = xln(1+x) - ∫xdln(x+1) = xln(1+x) - ∫xdx/(x+1) = xln(1+x) - ∫(x + 1 - 1)dx/(x+1) = xln(1+x) - ∫dx + ∫dx/(x+1) = xln(1+x) - x + ln(1+x) |¹₀ = ln2 - 1 + ln2 = ln4 – 1 = ln(4/e)

amanda2sempl: ∫dx/(1 + ∛1+x) = (1 + x = t³, dx = 3t²dt, x = 0 ⇒ t = 1, x = 7 ⇒ t = ∛8 = 2) = ∫3t²dt/(1+t) = 3∫(t² - 1 + 1)dt/(1+t) = 3∫(t - 1)dt + 3∫dt/(1+t) = 3·(0,5t² - t + ln|1+t|) |²₁ = 3·(0,5·2² - 2 + ln3 - 0,5 + 1 - ln2) = 3·(0,5 + ln1,5)