Primitive.

Definitie: Fie f, F : D R R doua functii, atunci F este o primitiva a lui f daca sinumai daca F (x) = f(x), x D. Multimea primitivelor functiei f se scrie:

f(x) dx = F (x) + C

1.

xr dx =

1

r + 1xr+1 + C, pentru orice r R \ {1}.

Consecinte:

r = 0,

1 dx = x+ C

r = 1,

x dx =

x2

2+ C

r = 2,

x2 dx =

x3

3+ C

r =1

2,

x dx =

2

3x

3

2 + C =2

3

x3 + C, x 0

r = 12,

1xdx = 2

x+ C

Pentru r = 1 avem:2.

1

xdx = ln |x|+ C, x R.

3.

ax dx =

1

ln aax + C.

ex dx = ex + C.

4.

sin x dx = cosx+ C.cosx dx = sin x+ C.

1

cos2 xdx = tg x+ C, x R \ {kpi + pi

2|k Z}.

1

sin2 xdx = ctg x+ C, x R \ {kpi|k Z}.

1

5.

1

x2 + a2dx =

1

aarctg

x

a+ C.

6.

1

x2 a2 dx =1

2aln

x ax+ a+ C, x R \ {a}.

7.

1

x2 + a2dx = ln(x+

x2 + a2) + C

1x2 a2 dx = ln |x+

x2 a2|+ C, x R \ [a, a].

8.

1

a2 x2 dx = arcsinx

a+ C, x (a, a).

9.

f (x) dx = f(x) + C.f(x) f (x) dx = 1

2f(x)2 + C.

f (x)

f(x)dx = ln |f(x)|+ C.

f (x) f(x)r dx = 1r + 1

f(x)r+1 + C, r 6= 1 .

10.

x

x2 + a2dx =

x2 + a2 + C

11.

x

a2 x2 dx = a2 x2 + C

12.

tg x dx = ln | cosx|+ C, x R \ {kpi + pi

2|k Z}.

13.

ctg x dx = ln | sin x|+ C, x R \ {kpi|k Z}.

2