Exan Primitive
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D´eterminer une primitive ......: (une primitive de f (x) est not´ee
R
f (x)dx
19.
R
(tan x + tan3 x)dx
20.
R
tan2 xdx 1 dx tan2 x
1.
R
(x + 1)(x2 + 2x − 1)4 dx
21.
R
1+
2.
R
1 − 2x dx (2x2 − 2x + 1)3
22.
R
b x dx(ind:´ecrire sous la forme :a+ ) x−1 x−1
3.
R
23.
R 4x + 5 b dx(ind:´ecrire sous la forme :a+ ) 2x + 1 2x + 1
4.
R
5.
R
√
1−x dx − 2x + 2
x2
(−2x + 1)5 dx
R 2x2 − 3x − 4 dx(ind:´ecrire sous la forme :ax + x−2 c b+ ) x−2 R 1 1 + dx 25. x−3 x+3 24.
2 dx 3x + 1
x2 dx x3 − 1 R cos x 7. dx sin x R 8. e2x+1 dx R 9. 2e−3x+2 dx 6.
R
26.
2 R 1 − x 10. e dx x2 11.
R
2x + 1 1 e x + 1 dx (x + 1)2
12.
R
sin x + x cos xdx
13.
R sin x − x cos x dx x2
R ln x − 1 dx x2 R√ x 15. x+1+ √ dx 2 x+1 14.
R ln x dx x R 1 17. dx x ln x 16.
18.
R ex − e−x dx ex + e−x 1
R (x + 3)2 dx (x + 2)2
Solution:
19.
R
(x + 1)(x2 + 2x − 1)4 dx =
2.
R
1 1 − 2x dx = 2 2 (2x2 − 2x + 1)3 4 (2x − 2x + 1)
3.
R
4.
R
5.
R
√
p 1−x dx = − (x2 − 2x + 2) − 2x + 2
x2
22.
R
x 1 x dx = x + ln |x − 1| ( =1+ ) x−1 x−1 x−1
ln |3x + 1|
R
11.
R
2x + 1 2x + 1 1 e x + 1 dx = e x + 1 (x + 1)2
12.
R
sin x
13.
R sin x − x cos x sin x dx = − x2 x
+
x cos xdx
R 4x + 5 dx = 2x + 2x + 1 3 ) 2x + 1
3 2
ln |2x + 1| (
4x + 5 = 2+ 2x + 1
R 2x2 − 3x − 4 dx = x2 + x − 2 ln |x − 2| x−2 2 2x2 − 3x − 4 = 2x + 1 − ) ( x−2 x−2 R 1 1 25. + dx = ln |(x − 3)(x + 3)| x−3 x+3 24.
2 2 R 1 − − x dx = 12 e x 10. e x2
26.
=
x sin x
R ln x − 1 1 dx = − ln x 2 x x R√ √ x 15. x+1+ √ dx = x x + 1 2 x+1 14.
R ln x ln x 1 dx = 12 ln2 x ( = ln x) x x x R 1 dx = ln |ln x| 17. x ln x
16.
18.
1 +
23.
3 x2 1 x − 1 dx = ln 3 x3 − 1 R cos x dx = ln |sin x| 7. sin x R 8. e2x+1 dx = 12 e2x+1 R 9. 2e−3x+2 dx = − 23 e−3x+2
6.
−1 (mettre au mˆeme tan x
1 dx = tan2 x d´enominateur)
R
6
2 3
tan2 x (penser `a mettre 2
21.
1 (−2x + 1)5 dx = − 12 (1 − 2x)
2 dx = 3x + 1
(tan x + tan3 x)dx =
tan x en facteur) R 20. tan2 xdx = tan x − x (tan2 x = 1 + tan2 x − 1)
1 2 (x + 2x − 1)5 10
1.
R
R ex − e−x dx = ln (ex + e−x ) ex + e−x 2
R (x + 3)2 1 dx = x − + 2 ln |x + 2| (x + 2)2 x+2 2 1 2 (x + 3) =1+ + ) ( 2 2 (x + 2) x+2 (x + 2)
R
f (x)dx
19.
R
(tan x + tan3 x)dx
20.
R
tan2 xdx 1 dx tan2 x
1.
R
(x + 1)(x2 + 2x − 1)4 dx
21.
R
1+
2.
R
1 − 2x dx (2x2 − 2x + 1)3
22.
R
b x dx(ind:´ecrire sous la forme :a+ ) x−1 x−1
3.
R
23.
R 4x + 5 b dx(ind:´ecrire sous la forme :a+ ) 2x + 1 2x + 1
4.
R
5.
R
√
1−x dx − 2x + 2
x2
(−2x + 1)5 dx
R 2x2 − 3x − 4 dx(ind:´ecrire sous la forme :ax + x−2 c b+ ) x−2 R 1 1 + dx 25. x−3 x+3 24.
2 dx 3x + 1
x2 dx x3 − 1 R cos x 7. dx sin x R 8. e2x+1 dx R 9. 2e−3x+2 dx 6.
R
26.
2 R 1 − x 10. e dx x2 11.
R
2x + 1 1 e x + 1 dx (x + 1)2
12.
R
sin x + x cos xdx
13.
R sin x − x cos x dx x2
R ln x − 1 dx x2 R√ x 15. x+1+ √ dx 2 x+1 14.
R ln x dx x R 1 17. dx x ln x 16.
18.
R ex − e−x dx ex + e−x 1
R (x + 3)2 dx (x + 2)2
Solution:
19.
R
(x + 1)(x2 + 2x − 1)4 dx =
2.
R
1 1 − 2x dx = 2 2 (2x2 − 2x + 1)3 4 (2x − 2x + 1)
3.
R
4.
R
5.
R
√
p 1−x dx = − (x2 − 2x + 2) − 2x + 2
x2
22.
R
x 1 x dx = x + ln |x − 1| ( =1+ ) x−1 x−1 x−1
ln |3x + 1|
R
11.
R
2x + 1 2x + 1 1 e x + 1 dx = e x + 1 (x + 1)2
12.
R
sin x
13.
R sin x − x cos x sin x dx = − x2 x
+
x cos xdx
R 4x + 5 dx = 2x + 2x + 1 3 ) 2x + 1
3 2
ln |2x + 1| (
4x + 5 = 2+ 2x + 1
R 2x2 − 3x − 4 dx = x2 + x − 2 ln |x − 2| x−2 2 2x2 − 3x − 4 = 2x + 1 − ) ( x−2 x−2 R 1 1 25. + dx = ln |(x − 3)(x + 3)| x−3 x+3 24.
2 2 R 1 − − x dx = 12 e x 10. e x2
26.
=
x sin x
R ln x − 1 1 dx = − ln x 2 x x R√ √ x 15. x+1+ √ dx = x x + 1 2 x+1 14.
R ln x ln x 1 dx = 12 ln2 x ( = ln x) x x x R 1 dx = ln |ln x| 17. x ln x
16.
18.
1 +
23.
3 x2 1 x − 1 dx = ln 3 x3 − 1 R cos x dx = ln |sin x| 7. sin x R 8. e2x+1 dx = 12 e2x+1 R 9. 2e−3x+2 dx = − 23 e−3x+2
6.
−1 (mettre au mˆeme tan x
1 dx = tan2 x d´enominateur)
R
6
2 3
tan2 x (penser `a mettre 2
21.
1 (−2x + 1)5 dx = − 12 (1 − 2x)
2 dx = 3x + 1
(tan x + tan3 x)dx =
tan x en facteur) R 20. tan2 xdx = tan x − x (tan2 x = 1 + tan2 x − 1)
1 2 (x + 2x − 1)5 10
1.
R
R ex − e−x dx = ln (ex + e−x ) ex + e−x 2
R (x + 3)2 1 dx = x − + 2 ln |x + 2| (x + 2)2 x+2 2 1 2 (x + 3) =1+ + ) ( 2 2 (x + 2) x+2 (x + 2)
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