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# 3.2.nenoteiktais integraalis

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### 3.2.nenoteiktais integraalis

1. 1. I. paņēmiens ax bI R x, n dx cx d n ax b dt bn t x n cx d a ct n dt b ax b n dx dt t a ct n cx d
2. 2. ax b n n t ax b cx d tcx d n n ax b cxt dt n n ax cxt dt b n n x a ct dt b n dt bx n a ct
3. 3. 2x 1 2x 1 t dx 2 x 2x 1 t 2 t 2 2t dt 2x t 1 2 2 tdt 2 t 1t 1 t 1 x 2 2 dx tdt
4. 4. 2 t 1 1 12 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
5. 5. II. paņēmiens 2 2 1 x a x dx x a2 x 2 2 a arcsin C 2 a 2 2 1x a dx x x2 a2 a 2 ln x x2 a2 C 2 2 2 1 2 2 2 2 2x a dx x x a a ln x x a C 2
6. 6. 2 x 4 x 13dx 2 x 2 9dx1 2 9 x 2 x 4 x 13 ln x 2 x 2 4 x 13 C2 2
7. 7. III. paņēmiens 2aN 2ax b b Mx N M M dx dxax 2 bx c 2a ax 2 bx c 2 d ax bx c 2ax b dxM d ax2 bx c bM dx N2a 2 ax bx c 2a ax 2 bx c
8. 8. 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
9. 9. III. paņēmiens otrais gadījumsM bM dx 2 ax 2 bx c N2a 2a X 2 A 2M 2 bM 2 2 2 ax bx c N ln x X A C2a 2a
10. 10. 2x 3 2x 2 1 dx dx8 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
11. 11. IV. paņēmiens Pn x dx 2 ax bx c 2 dxQn 1 x ax bx c k 2 ax bx c
12. 12. 2 x 2 5x 23 2 dx dx Ax B x 2 x 17 k x 2 2 x 17 x 2 2 x 17Atvasina vienādojuma abas puses2 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 17Atbrīvojas no saknes 2 x 2 5 x 23 A x2 2 x 17 Ax 2 Ax Bx B k
13. 13. 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 517 A B k 23 17 B k 23 2A 2 A 1 3A B 5 B 217 A B k 23 17 2 k 23
14. 14. A 1 B 2 k 42 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
15. 15. V. paņēmiensI1 R u, m 2 u 2 du u m sin t vai u m cost 2 2 m mI2 R u, u m du u vai u sin t costI3 R u, u 2 m 2 du u m tgt vai u m ctgt
16. 16. 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 2dx 2 costdt
17. 17. dx 2 costdt 1 dt 4 x 2 3 8 cos3 t 4 cos2 t x1 1 sin t 1 2 tgt C C C4 4 cost 4 1 4 x2 2 1 x C 4 4 x 2