2. I. paņēmiens
ax b
I R x, n dx
cx d
n
ax b dt b
n t x n
cx d a ct
n
dt b
ax b n dx dt
t a ct n
cx d

3. ax b n n
t ax b cx d t
cx d
n n
ax b cxt dt
n n
ax cxt dt b
n n
x a ct dt b
n
dt b
x n
a ct

4. 2x 1 2x 1 t
dx 2
x 2x 1 t
2
t 2
2t dt 2x t 1
2
2
tdt 2 t 1
t 1 t 1 x
2
2
dx tdt

5. 2
t 1 1 1
2 2
dt 2 dt 2 2
dt
t 1 t 1
t 1
2t ln C
t 1
2x 1 1
2 2 x 1 ln C
2x 1 1

6. II. paņēmiens
2 2 1 x
a x dx x a2 x 2 2
a arcsin C
2 a
2 2 1
x a dx x x2 a2 a 2 ln x x2 a2 C
2
2 2 1 2 2 2 2 2
x a dx x x a a ln x x a C
2

7. 2
x 4 x 13dx
2
x 2 9dx
1 2 9
x 2 x 4 x 13 ln x 2 x 2 4 x 13 C
2 2

8. III. paņēmiens
2aN
2ax b b
Mx N M M
dx dx
ax 2 bx c 2a ax 2 bx c
2
d ax bx c 2ax b dx
M d ax2 bx c bM dx
N
2a 2
ax bx c 2a ax 2
bx c

9. III. paņēmiens pirmais gadījums
M 2 bM dx
2 ax bx c N
2a 2a A2 X2
M bM X
2 ax 2 bx c N arcsin C
2a 2a A

10. III. paņēmiens otrais gadījums
M bM dx
2 ax 2 bx c N
2a 2a X 2
A 2
M 2 bM 2 2
2 ax bx c N ln x X A C
2a 2a

11. 2x 3 2x 2 1
dx dx
8 2x x2 8 2x x2
2
d 8 2x x 2 x 2 dx
d (8 2 x x 2 ) d x 1
dx 2
2
8 2x x 9 x 1
2 x 1
2 8 2x x arcsin C
3

12. IV. paņēmiens
Pn x dx
2
ax bx c
2 dx
Qn 1 x ax bx c k
2
ax bx c

13. 2 x 2 5x 23 2 dx
dx Ax B x 2 x 17 k
x 2 2 x 17 x 2 2 x 17
Atvasina vienādojuma abas puses
2 x 2 5x 23 2 Ax B x 1 k
A x 2 x 17
2
x 2 x 17 x 2 2 x 17 x 2 2 x 17
Atbrīvojas no saknes
2 x 2 5 x 23 A x2 2 x 17 Ax 2 Ax Bx B k

14. 2 x 2 5x 23 Ax2 2 Ax 17 A Ax2 Ax Bx B k
A A 2 A 1
2A A B 5 3 B 5
17 A B k 23 17 B k 23
2A 2 A 1
3A B 5 B 2
17 A B k 23 17 2 k 23

15. A 1
B 2
k 4
2 x 2 5x 23 2 dx
dx x 2 x 2 x 17 4
2
x 2 x 17 x 2 2 x 17
2 d x 1
x 2 x 2 x 17 4
2
x 1 16
x 2 x 2 2x 17 4 ln x 1 x 2 2x 17 C

16. V. paņēmiens
I1 R u, m 2 u 2 du u m sin t vai u m cost
2 2 m m
I2 R u, u m du u vai u
sin t cost
I3 R u, u 2 m 2 du u m tgt vai u m ctgt

17. dx
x 2 sin t
2 3
4 x
x
sin t
2
2 3 2 3
4 x 4 2 sin t cost 1 sin 2 t
2
2 3 2 3 x
4 4 sin t 4 cos t cost 1
2
3 3
2 cos t 8 cos t
1 2
cost 4 x
2
dx 2 costdt

18. dx 2 costdt 1 dt
4 x 2 3 8 cos3 t 4 cos2 t
x
1 1 sin t 1 2
tgt C C C
4 4 cost 4 1 4 x2
2
1 x
C
4 4 x 2