Matemática 1
Anatolie Sochirca ACM DEETC ISEL
Integral definido. Exercícios resolvidos. a) Calcular os integrais definidos utilizando a fórmula de Barrow.
1
∫
Exercício 1.
1 + x dx .
0
1
1 1 1 (1 + x) 2 +1 1 + x d (1 + x) = ∫ (1 + x) 2 d (1 + x) = 1 0 + 1 2
1
∫
1 + x dx = ∫
0
0
3 2 = ⋅ (1 + x ) 2 3
1
0
(
1
∫ (x
−1
∫ (x
−1
x dx 2
)
+1
x dx 2
)
+1
2
1
= 0
.
1 d (x2 ) 1 1 1 d (x2 ) d ( x 2 + 1) 1 1 1 =∫ 2 = ⋅ = ⋅ = ⋅ x2 +1 ∫ ∫ ∫ 2 2 2 2 2 2 2 −1 x + 1 2 −1 x + 1 2 −1 −1 x + 1
(
)
(
1
2
0
3 (1 + x) 2 = 3 2
3 3 2 32 2 2 2 2 = ⋅ (1 + 1) − (1 + 0) = ⋅ 2 − 1 = ⋅ 2 2 − 1 . 3 3 3
Exercício 2.
1
1
(
)
)
1
− 2 +1 1 x 2 + 1 = ⋅ 2 − 2 + 1
−1
(
)
((
)
1 = − ⋅ x2 +1 2
−1
(
)
1 −1
)
1 1 = − ⋅ 2 2 x + 1
1
−1
)
−2
d ( x 2 + 1) =
1 1 1 =0 = − ⋅ 2 − 2 2 1 + 1 (−1) + 1
(1 + l nx) dx . x 1
e
Exercício 3.
∫
(1 + l nx) dx dx =∫ (1 + l nx) ⋅ = ∫ (1 + l nx) ⋅ d (l nx) = ∫ d (l nx) + ∫ l nx d (l nx) = ∫1 x x 1 1 1 1 e
e
= (l nx ) 1
e
e
e
e
e
ln2 x l n 2e l n 21 1 3 = (l n e − l n1) + = 1 + = . + − 2 2 2 2 1 2
1