Question

Вычислите определённый интеграл:
$\int\limits^3_1 {x^3} \, dx$

$\int\limits^2_1 ({4^{5}-3x^{4}+3^{3}-1}) \, dx$

Answer 1

[1;3] ∫ x^3 dx = x^4/4 [1;3] = 1/4 * ( 3^4 -1^4 )= 1/4 (81 -1) = 20
[1;2] ∫ (x^5 -3x^4 +x^3 -1) dx = (x^6/6 -3*x^5/5 + x^4/4 - x) [1;2] =
       = 1/6 (2^6 -1^6) -3/5 (2^5 -1^5) +1/4 (2^4 -1^4) - (2-1) =
= 1/6 (2^6 -1^6) -3/5 (2^5 -1^5) +1/4 (2^4 -1^4) - (2-1) =
= 21/2 - 93/5 + 15/4 -1 = -107/20 = -5,35