log x(log(log x

Esercizio 1

Calcolare i seguenti integrali indefiniti:

(a)

Z x²+ 1 x√

x dx;  2x²− 6 3√

x + c



(b)

Z x⁵− x³

√x²− 1dx;  1

5(x²− 1)^5/2+ (x²− 1)^3/2+ (x²− 1)^1/2+ c



(c)

Z x²+ x + 1 p(1 − x)³dx;



−2

3(1 − x)^3/2+ 6(1 − x)^1/2+ 6(1 − x)^−1/2+ c



(d)

Z log(log x)

x dx; 

log x(log(log x) − 1) + c

(e) Z

x⁵e^−x2dx; −e^−x2

2 (x⁴ + 2x²+ 2) + c

!

(f) Z

xe^√xdx; 

−2e^√x(3x − 6x^1/2− x^3/2+ 6) + c Esercizio 2

Determinare la primitiva di f (x) passante per il punto P . (a) f (x) = log(2x + 1), P = (0, −1)

((x + 1/2) log(2x + 1) − x − 1) (b) f (x) = 3 − 4x, P = (1, 6)

(3x − 2x²+ 5)

(c) f (x) = ax²+ bx + c, P = (1, 0)

a

3(x³− 1) + ^b2(x²− 1) + c(x − 1)
Esercizio 3

Determinare f sapendo che (a) f^′(x) = 2x − 1, f(3) = 4.

(f (x) = x²− x − 2)

(b) f^′′(x) = x²− x, f^′(1) = 0, f (1) = 2.

f (x) = ^x₁₂⁴ − ^x6³ +^x₆ + ²³₂₄ (c) f^′′(x) = e^2x, f^′(0) = f (0) = 1.

f (x) = ^e2x₄ +^x₂ + ³₄ Esercizio 4

Calcolare i seguenti integrali definiti:

1

(a) Z ^e

1

log x

√x dx; 4 − 2√ e

(b) Z ⁴

0

e^√x

√xdx; 

2 e²− 1


(c) Z ⁴

0

√xe^√xdx; 

4 e²− 1


(d)

Z ^√27

√6

x√

x²− 2dx;  39

(e) Z ¹

0

x(x²+ 1)³dx; 15 8



(f) Z ⁸

1

1 + x^1/3
2

x^2/3 dx;  19 Esercizio 5

Determinare l’area della regione piana (a) sopra y = x/2 e sotto y =√

x. (4/3) (b) sopra y = |x| e sotto y = 12 − x². (45)

(c) limitata dall’iperbole xy+a² = 0 e dalla retta 2x−y+3a = 0. (a²(3/4 − log 2))

2