C, Z 1 1 + x2dx = arctan(x

Corso di laurea in Geologia Istituzioni di matematiche

Esercizi n. 1617/2/7 Regole di integrazione:

Z

(f (x) + g(x)) dx = Z

f (x) dx + Z

g(x) dx, Z

a · f (x) dx = a · Z

f (x) dx, Z

xⁿdx = 1

n + 1xⁿ⁺¹+ C, Z

sin(x) dx = − cos(x) + C, Z

cos(x) dx = sin(x) + C,

Z 1

cos²(x)dx = tan(x) + C,

Z 1

xdx = log(|x|) + C, Z

e^xdx = e^x+ C, Z

log(x) dx = xlog(x) − x + C, Z

x^adx = 1

a + 1x^a+1+ C, a 6= −1,

Z √

x dx = 2 3x√

x + C,

Z 1

√1 − x²dx = arcsin(x) + C,

Z 1

1 + x²dx = arctan(x) + C, Z f⁰(x)

f (x) dx = log(|f (x)|) + C, Z

f (x)ⁿ· f⁰(x) dx = 1

n + 1f (x)ⁿ⁺¹+ C Formula di integrazione per parti:

Z

f⁰(x) · g(x) dx = f (x) · g(x) − Z

f (x) · g⁰(x) dx

Z

f (g(x)) · g⁰(x) dx = F (g(x)) + c (dove F (x) `e una primitiva di f (x)).

1

1. Integrali indefiniti (immediati dalle formule):

Z

8 x³+ 6 x, Z

(sin(x) + 2 cos(x)) dx Z 1

x²dx, Z 4

x³dx.

Risp.:

2 x⁴+ 3 x²+ c, − cos(x) + 2 sin(x) + c, −1

x+ c, −2 x²+ c.

2. Integrali indefiniti (immediati dalle formule):

Z 3

cos²(x)dx, Z √

x³dx,

Z 1

√

1 − x²dx, Risp.:

3 tan(x) + c, 2 5x²√

x + c, arcsin (x) + c 3. FormulaR (f⁰(x)/f (x)) dx = log(|f (x)|) + c:

Z 3x

x²+ 8dx

Z sin(x) cos(x) + 2dx

Z 1 + sin(x) x − cos(x)dx.

Risp.:

3

2 log x²+ 8
+ c, − log (cos (x) + 2) + c, log |x − cos (x)| + c 4. Integrazione per parti:

Z

2x sin(x) dx, Z

x log(x) dx, Z

x²e^xdx.

Risp.:

−2 x cos (x) + 2 sin (x) + c, 1

2x²log (x) −1

4x²+ c, x²− 2 x + 2
e^x+ c.

5. FormulaR f (g(x)) · g⁰(x) dx = F (g(x)) + c:

Z

sin(4x) dx, Z

x²cos(x³) dx, Z

x log(x²+ 2) dx

Z 2x

1 + x⁴dx,

Z 2x

√1 − x⁴dx Z

xp

x²+ 2 dx.

Risp.:

−1

4 cos (4 x) + c, 1

3 sin x³
+ c, −1 2x²+1

2 x²+ 2
log x²+ 2
− 1 + c arctan(x²) + c, arcsin(x²) + c, 1

3 q

(x²+ 2)³+ c.

2

6. Miscellanea:

Z x + 3 x + 5dx,

Z

(x + 2) sin(3x) dx, Z

(x − 1)e^2x+3dx

Risp.:

x−2 log (x + 5)+c, −1

3x cos (3 x)−2

3 cos (3 x)+1

9 sin (3 x)+c, 1

4(2 x − 3)e^{(2 x+3)}+c.

3