微積分常用公式

1、1.常用等價無窮小當 時 x→0 x~sin x~tan x~arcsinx~arctanx~ln (1 + x)~ax ? 1ln a ~(1 + x)b ? 1b (其中a > 0,𝑏 ≠ 0)x~ex ? 112x2~1 ? cos x1nx~n 1 + x ? 1 α~(1 + x)α2.常用極限 1. 2. lim x→∞nkan = 0 ,(a > 1) lim x→∞cnn! =

2、 0,(c > 0)3. 4. lim x→∞nqn,(|q| 0)5. 6. lim x→∞n n = 1 lim x→∞loga nn = 0,(a > 1)7. 8. lim x→∞1n n! = 0 lim x→∞(1 +1n)n = e9. 10. lim x→∞nn n! = e lim x→∞sin xx = 011. 12. limx→ + ∞loga xxε = 0

3、 ,(a > 1,𝜀 > 0) lim x→∞(1p + 2p + … + npnp + 1 ) =1p + 113. 14. lim x→∞(1p + 2p + … + npnp + 1 ?np + 1) =12 lim x→∞(1p + 3p + … + (2n ? 1)pnp + 1 ) =2pp + 115. limx→∞(1n + 1 +1n + 2 + … +12n) = ln 216.

4、 17. lim x→0sin xx = 1 lim x→0(1 + x)1x = e18. 19. lim x→0ax ? 1x = ln a lim x→0(1 + a)μ ? 1a = μ20. 21. lim x→0ln (1 + x)x = 1 lim x→0arcsinxx = 122. 23. lim x→0arctanxx = 1 lim x→0(1 + mx)n ? (1 + nx)mx2 =12m

5、n(n ? m)24. 25. lim x→0m 1 + αx ?n 1 + βxx =αm ?βn,(mn ≠ 0) lim x→0m 1 + αx ?n 1 + βx ? 1xαm +βn，（mn ≠ 0）26. 27. lim x→1xm ? 1xn ? 1 =mn,(m,n為自然數(shù)) lim x→1(m1 ? xm ?n1 ? xn) =m ? n228. lim x→1m x ? 1n x ? 1 =nm,(m,n ∈

6、 Z)29. 若 Xn（n=1,2…）收斂,則算數(shù)平均值的序列 也收斂，且 ζ𝑛 =1n(X1 + X2 + …Xn),(n = 1,2?)lim x→∞x1 + x2 + … + xnn = limx→∞xn30. 若序列 Xn(n=1,2…)收斂，且 Xn>0,則limx→∞n x1 + x2 + … + xn = limx→∞Xn11. 12. y = arcsinx dy =11 ? x2dx y

7、= arccosx dy =?11 ? x2dx13. 14. y = arctanx dy =11 + x2dx y = arccotx dy =?11 + x2dx15. 16. y = shx dy = chxdx y = chx dy = shxdx17. 18. y = thx dy =1ch2 xdx y = cthx

8、 dy =?1sh2 xdx6.不定積分表1. 2. ∫0dx = c ∫1dx = x + c3. 4. . ∫xμdx =1μ + 1xμ + 1 + c ∫1xdx = ln |x| + c5. 6. ∫11 + x2dx = arctanx + c ∫11 ? x2dx = arcsinx + c7. 8. ∫axdx =axln a + c ∫sin xdx =? cos x

9、+ c9. 10. ∫cos xdx = sin x + c ∫1sin2 xdx =? cot x + c11. 12. ∫1cos2 xdx = tan x + c ∫sh xdx = ch x + c13. 14. ∫ch xdx = sh x + c ∫1sh2 xdx = ? cth x + c15. 16 ∫1ch2 xdx = th x + c ∫dxax + x2 =1aarctanxa + c

10、,(a ≠ 0)17. 18. ∫dxax ? x2 =12aln |a + xa ? x| + c ∫xdxax ± x2 =±12ln |a2 ± x2| + c19. 20. ∫dxa2 ? x2 = arcsinxa + c,(a > 0) ∫dxx2 ± a2 = ln |x + x2 ± a2| + c21. ∫xdxa2 ± x2 =± a

11、2 ± x2 + c22. ∫ a2 ? x2dx =x2 a2 ? x2 + arcsinxa + c,(a > 0)22. ∫ x2 ± a2dx =x2 x2 ± a2 ±a22ln |x + x2 ± a2| + c7.三角學公式 sin2 θ cos2 θ1.基本關系1. 2. sin θ ? csc θ = 1 cos θ ? sec θ = 13.