calculo

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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

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