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文章目录
- 含有 a x + b ax+b ax+b的积分
- 含有 a x + b \sqrt{ax+b} ax+b 的积分
- 含有 x 2 ± a x^2 \pm a x2±a 的积分
- 含有 a x 2 + b ax^2 +b ax2+b 的积分
- 含有 a x 2 + b x + c ax^2+bx+c ax2+bx+c 的积分
- 含有 x 2 + a 2 ( a > 0 ) \sqrt{x^2+a^2} \quad (a>0) x2+a2(a>0) 的积分
- 含有 x 2 − a 2 ( a > 0 ) \sqrt{x^2-a^2} \quad (a>0) x2−a2(a>0) 的积分
- 含有 a 2 − x 2 ( a > 0 ) \sqrt{a^2-x^2} \quad (a>0) a2−x2(a>0) 的积分
- 含有 ± a x 2 + b x + c ( a > 0 ) \sqrt{\pm ax^2+bx+c} \quad (a>0) ±ax2+bx+c(a>0)的积分
含有 a x + b ax+b ax+b的积分
1. ∫ d x a x + b = 1 a ln ∣ a x + b ∣ + C \begin{equation} 1.\,\int\!\! \frac{dx}{ax+b}=\frac{1}{a}\ln \vert ax+b \vert +C \end{equation} 1.∫ax+bdx=a1ln∣ax+b∣+C
2. ∫ ( a x + b ) μ d x = 1 a ( μ + 1 ) ( a x + b ) μ + 1 + C ( μ ≠ − 1 ) \begin{equation} 2.\,\int\!\! (ax+b)^\mu dx=\frac{1}{a(\mu+1)}(ax+b)^{\mu+1}+C \qquad (\mu \neq -1) \end{equation} 2.∫(ax+b)μdx=a(μ+1)1(ax+b)μ+1+C(μ=−1)
3. ∫ x a x + b d x = 1 a 2 ( a x + b − b ln ∣ ( x + b ) ∣ ) + C \begin{equation} 3.\,\int\!\! \frac{x}{ax+b}dx=\frac{1}{a^2}(ax+b-b\ln\vert (x+b) \vert ) +C \end{equation} 3.∫ax+bxdx=a21(ax+b−bln∣(x+b)∣)+C
4. ∫ x 2 a x + b d x = 1 a 3 [ 1 2 ( a x + b ) 2 − 2 b ( a x + b ) + b 2 ln ∣ a x + b ∣ ] + C \begin{equation} 4.\,\int\!\! \frac{x^2}{ax+b}dx=\frac{1}{a^3}\Big[\frac{1}{2}(ax+b)^2-2b(ax+b)+b^2\ln\vert ax+b\vert\Big]+C \end{equation} 4.∫ax+bx2dx=a31[21(ax+b)2−2b(ax+b)+b2ln∣ax+b∣]+C
5. ∫ d x x ( a x + b ) = − 1 b ln ∣ a x + b x ∣ + C \begin{equation} 5.\,\int\!\! \frac{dx}{x(ax+b)}=-\frac{1}{b}\ln \Big\vert \frac{ax+b}{x}\Big\vert +C \end{equation} 5.∫x(ax+b)dx=−b1ln xax+b +C
6. ∫ d x x 2 ( a x + b ) = − 1 b x + a b 2 ln ∣ a x + b x ∣ + C \begin{equation} 6.\,\int\!\! \frac{dx}{x^2(ax+b)}=-\frac{1}{bx}+\frac{a}{b^2}\ln\Big\vert\frac{ax+b}{x}\Big\vert+C \end{equation} 6.∫x2(ax+b)dx=−bx1+b2aln xax+b +C
7. ∫ x ( a x + b ) 2 d x = 1 a 2 ( ln ∣ a x + b ∣ + b a x + b ) + C \begin{equation} 7.\,\int\!\! \frac{x}{(ax+b)^2}dx=\frac{1}{a^2}\Big(\ln \vert ax+b \vert + \frac{b}{ax+b}\Big)+C \end{equation} 7.∫(ax+b)2xdx=a21(ln∣ax+b∣+ax+bb)+C
8. ∫ x 2 ( a x + b ) 2 d x = 1 a 3 ( a x + b − 2 b ln ∣ a x + b ∣ − b 2 a x + b ) + C \begin{equation} 8.\,\int\!\! \frac{x^2}{(ax+b)^2}dx= \frac{1}{a^3}\Big(ax+b-2b\ln\vert ax+b \vert - \frac{b^2}{ax+b}\Big ) + C \end{equation} 8.∫(ax+b)2x2dx=a31(ax+b−2bln∣ax+b∣−ax+bb2)+C
9. ∫ d x x ( a + b ) 2 = 1 b ( a x + b ) − 1 b 2 ln ∣ a x + b x ∣ + C \begin{equation} 9.\,\int\!\! \frac{dx}{x(a+b)^2}=\frac {1}{b(ax+b)}-\frac{1}{b^2}\ln\Big\vert \frac{ax+b}{x}\Big\vert + C \end{equation} 9.∫x(a+b)2dx=b(ax+b)1−b21ln xax+b +C
含有 a x + b \sqrt{ax+b} ax+b 的积分
10. ∫ a x + b d x = 2 3 a ( a x + b ) 3 + C \begin{equation} 10.\,\int\!\! \sqrt{ax+b} dx= \frac{2}{3a}\sqrt{(ax+b)^3}+C \end{equation} 10.∫ax+bdx=3a2(ax+b)3+C
11. ∫ x a x + b d x = 2 15 a 2 ( 3 a x − 2 b ) ( a x + b ) 3 + C \begin{equation} 11.\,\int\!\! x \sqrt{ax+b}dx=\frac{2}{15a^2}(3ax-2b)\sqrt{(ax+b)^3}+C \end{equation} 11.∫xax+bdx=15a22(3ax−2b)(ax+b)3+C
12. ∫ x 2 a x + b d x = 2 105 a 3 ( 15 a 2 x 2 − 12 a b x + 8 b 2 ) ( a x + b ) 3 + C \begin{equation} 12.\,\int\!\! x^2 \sqrt{ax+b}dx=\frac{2}{105a^3}(15a^2x^2-12abx+8b^2)\sqrt{(ax+b)^3}+C \end{equation} 12.∫x2ax+bdx=105a32(15a2x2−12abx+8b2)(ax+b)3+C
13. ∫ x a x + b d x = 2 3 a 2 ( a x − 2 b ) a x + b + C \begin{equation} 13.\,\int\!\! \frac{x}{\sqrt {ax+b}}dx= \frac{2}{3a^2}(ax-2b)\sqrt{ax+b}+C \end{equation} 13.∫ax+bxdx=3a22(ax−2b)ax+b+C
14. ∫ x 2 a x + b d x = 2 15 a 2 ( 2 a 2 x 2 − 4 a b x + 8 b 2 ) a x + b + C \begin{equation} 14.\,\int\!\! \frac{x^2}{\sqrt {ax+b}}dx= \frac{2}{15a^2}(2a^2x^2-4abx+8b^2)\sqrt{ax+b}+C \end{equation} 14.∫ax+bx2dx=15a22(2a2x2−4abx+8b2)ax+b+C
15. ∫ d x x a x + b = { 1 b ln ∣ a x + b − b a x + b + b ∣ + C ( b > 0 ) 2 − − b arctan a x + b − b + C ( b < 0 ) \begin{equation} 15.\,\int\!\! \frac{dx}{x\sqrt{ax+b}}=\left \{ \begin{array}{cc} \!\!\!\!\frac{1}{\sqrt{b}}\ln \Big \vert \frac{ \sqrt{ax+b} - \sqrt{b}}{\sqrt{ax+b}+\sqrt{b}} \Big \vert + C & \textrm ( b>0) \\ \frac{2}{-\sqrt{-b}}\arctan \sqrt{\frac{ax+b}{-b}}+C & \textrm (b<0) \end{array} \right. \end{equation} 15.∫xax+bdx=⎩ ⎨ ⎧b1ln ax+b+bax+b−b +C−−b2arctan−bax+b+C(b>0)(b<0)
16. ∫ d x x 2 a x + b = − a x + b b x − a 2 b ∫ d x x a x + b \begin{equation} 16.\,\int\!\! \frac{dx}{x^2\sqrt{ax+b}} = - \frac{\sqrt{ax+b}}{bx} - \frac{a}{2b}\int\!\! \frac{dx}{x\sqrt{ax+b}} \end{equation} 16.∫x2ax+bdx=−bxax+b−2ba∫xax+bdx
17. ∫ a x + b x d x = 2 a x + b + b ∫ d x x a x + b \begin{equation} 17.\,\int\!\! \frac{\sqrt{ax+b}}{x}dx = 2 \sqrt{ax+b}+b \int\!\! \frac{dx}{x\sqrt{ax+b}} \end{equation} 17.∫xax+bdx=2ax+b+b∫xax+bdx
18. ∫ a x + b x 2 d x = − a x + b x + a 2 ∫ d x x a x + b \begin{equation} 18.\,\int\!\! \frac{\sqrt{ax+b}}{x^2}dx= - \frac{\sqrt{ax+b}}{x}+ \frac{a}{2}\int\!\! \frac{dx}{x\sqrt{ax+b}} \end{equation} 18.∫x2ax+bdx=−xax+b+2a∫xax+bdx
含有 x 2 ± a x^2 \pm a x2±a 的积分
19. ∫ d x x 2 + a 2 = 1 a arctan x a + C \begin{equation} 19.\,\int\!\! \frac{dx}{x^2+a^2}= \frac{1}{a} \arctan \frac{x}{a} +C \end{equation} 19.∫x2+a2dx=a1arctanax+C
20. ∫ d x ( x 2 + a 2 ) n = x 2 ( n − 1 ) a 2 ( x 2 + a 2 ) n − 1 + 2 n − 3 2 ( n − 1 ) a 2 ∫ d x ( x 2 + a 2 ) n − 1 \begin{equation} 20.\,\int\!\! \frac{dx}{(x^2+a^2)^n} = \frac{x}{2(n-1)a^2(x^2+a^2)^{n-1}} + \frac {2n-3}{2(n-1)a^2} \int\!\! \frac {dx}{(x^2+ a^2)^{n-1}} \end{equation} 20.∫(x2+a2)ndx=2(n−1)a2(x2+a2)n−1x+2(n−1)a22n−3∫(x2+a2)n−1dx
21. ∫ d x x 2 − a 2 = 1 2 a ln ∣ x − a x + a ∣ + C \begin{equation} 21.\,\int\!\!\ \frac{dx}{x^2-a^2}= \frac{1}{2a} \ln \Big \vert \frac{x-a}{x+a}\Big \vert + C \end{equation} 21.∫ x2−a2dx=2a1ln x+ax−a +C
含有 a x 2 + b ax^2 +b ax2+b 的积分
22. ∫ d x a x 2 + b = { 1 a b arctan a b x + C ( b > 0 ) 1 2 − a b ln ∣ a x − − b a x + − b ∣ + C ( b < 0 ) \begin{equation} 22.\,\int\!\! \frac{dx}{ax^2+b} = \left \{ \begin{array}{cc} \!\!\!\!\frac{1}{\sqrt{ab}}\arctan \sqrt {\frac {a}{b}}x +C & \textrm (b>0) \\ \frac{1}{2 \sqrt{-ab}} \ln\Big \vert \frac { \sqrt {a}x - \sqrt {-b}}{\sqrt{a}x+ \sqrt{-b}} \Big \vert + C & \textrm (b<0) \end{array} \right. \end{equation} 22.∫ax2+bdx={ab1arctanbax+C2−ab1ln ax+−bax−−b +C(b>0)(b<0)
23. ∫ x a x 2 + b d x = 1 2 a ln ∣ a x 2 + b ∣ + C \begin{equation} 23.\,\int\!\! \frac{x} {ax^2 +b} dx = \frac{1}{2a} \ln \vert ax^2+b \vert +C \end{equation} 23.∫ax2+bxdx=2a1ln∣ax2+b∣+C
24. ∫ x 2 a x 2 + b d x = x a − b a ∫ d x a x 2 + b \begin{equation} 24.\,\int\!\! \frac{x^2}{ax^2+b} dx= \frac{x}{a} - \frac{b}{a} \int\!\! \frac{dx}{ax^2+b} \end{equation} 24.∫ax2+bx2dx=ax−ab∫ax2+bdx
25. ∫ d x x ( a x 2 + b ) = 1 2 b ln x 2 ∣ a x 2 + b ∣ + C \begin{equation} 25.\,\int\!\! \frac {dx} { x(ax^2+b)} = \frac {1}{2b} \ln \frac {x^2}{ \vert ax^2+b \vert } +C \end{equation} 25.∫x(ax2+b)dx=2b1ln∣ax2+b∣x2+C
26. ∫ d x x 2 ( a x 2 + b ) = − 1 b x − a b ∫ d x a x 2 + b \begin{equation} 26.\,\int\!\! \frac {dx}{x^2(ax^2+b)} = - \frac{1}{bx} - \frac{a}{b} \int\!\! \frac {dx}{ax^2+b} \end{equation} 26.∫x2(ax2+b)dx=−bx1−ba∫ax2+bdx
27. ∫ d x x 3 ( a x 2 + b ) = a 2 b 2 ln ∣ a x 2 + b ∣ x 2 − 1 2 b x 2 + C \begin{equation} 27.\,\int\!\! \frac{dx}{x^3(ax^2+b)} = \frac {a}{2b^2} \ln \frac { \vert ax^2+b\vert }{x^2} - \frac{1}{2bx^2} + C \end{equation} 27.∫x3(ax2+b)dx=2b2alnx2∣ax2+b∣−2bx21+C
28. ∫ d x ( a x 2 + b ) 2 = x 2 b ( a x 2 + b ) + 1 2 b ∫ d x a x 2 + b \begin{equation} 28.\,\int\!\! \frac{dx}{(ax^2+b)^2} = \frac{x}{2b(ax^2+b)} + \frac{1}{2b} \int\!\! \frac {dx} { ax^2+b} \end{equation} 28.∫(ax2+b)2dx=2b(ax2+b)x+2b1∫ax2+bdx
含有 a x 2 + b x + c ax^2+bx+c ax2+bx+c 的积分
29. ∫ d x a x 2 + b x + c d x = { 2 4 a c − b 2 arctan 2 a x + b 4 a c − b 2 + C ( b 2 < 4 a c ) 1 b 2 − 4 a c ln ∣ 2 a x + b − b 2 − 4 a c 2 a x + b + b 2 − 4 a c ∣ + C ( b 2 > 4 a c ) \begin{equation} 29.\,\int\!\! \frac{dx}{ax^2+bx+c}dx= \left \{ \begin{array}{cc} \!\!\! \frac{2}{\sqrt{4ac-b^2}} \arctan \frac {2ax+b}{\sqrt{4ac-b^2}}+C & \textrm (b^2<4ac) \\ \frac {1}{\sqrt {b^2-4ac}} \ln \Big \vert \frac {2ax+b-\sqrt{b^2-4ac}} {2ax+b+ \sqrt{b^2-4ac}} \Big \vert +C & \textrm (b^2>4ac) \end{array} \right. \end{equation} 29.∫ax2+bx+cdxdx={4ac−b22arctan4ac−b22ax+b+Cb2−4ac1ln 2ax+b+b2−4ac2ax+b−b2−4ac +C(b2<4ac)(b2>4ac)
30. ∫ x a x 2 + b x + c d x = 1 2 a ln ∣ a x 2 + b x + c ∣ − b 2 a ∫ d x a x 2 + b x + c \begin{equation} 30.\,\int\!\! \frac{x}{ax^2+bx+c} dx = \frac{1}{2a}\ln \vert ax^2+bx+c \vert - \frac{b}{2a} \int\!\! \frac {dx}{ax^2+bx+c} \end{equation} 30.∫ax2+bx+cxdx=2a1ln∣ax2+bx+c∣−2ab∫ax2+bx+cdx
含有 x 2 + a 2 ( a > 0 ) \sqrt{x^2+a^2} \quad (a>0) x2+a2(a>0) 的积分
31. ∫ d x x 2 + a 2 = a r s h x a + C 1 = ln ( x + x 2 + a 2 ) + C \begin{equation} 31.\,\int\!\! \frac{dx}{\sqrt{x^2+a^2}} = arsh \frac{x}{a}+C1= \ln (x+\sqrt{x^2+a^2})+C \end{equation} 31.∫x2+a2dx=arshax+C1=ln(x+x2+a2)+C
32. ∫ d x ( x 2 + a 2 ) 3 = x a 2 x 2 + a 2 + C \begin{equation} 32.\,\int\!\! \frac{dx}{\sqrt{(x^2+a^2)^3}}= \frac{x}{a^2\sqrt{x^2+a^2}}+C \end{equation} 32.∫(x2+a2)3dx=a2x2+a2x+C
33. ∫ x x 2 + a 2 d x = x 2 + a 2 + C \begin{equation} 33.\,\int\!\! \frac{x}{\sqrt{x^2+a^2}}dx=\sqrt{x^2+a^2}+C \end{equation} 33.∫x2+a2xdx=x2+a2+C
34. ∫ x ( x 2 + a 2 ) 3 d x = − 1 x 2 + a 2 + C \begin{equation} 34.\,\int\!\! \frac{x}{\sqrt{(x^2+a^2)^3}}dx= - \frac{1}{\sqrt{x^2+a^2}}+C \end{equation} 34.∫(x2+a2)3xdx=−x2+a21+C
35. ∫ x 2 x 2 + a 2 d x = x 2 x 2 + a 2 − a 2 2 ln ( x + x 2 + a 2 ) + C \begin{equation} 35.\,\int\!\! \frac{x^2}{\sqrt{x^2+a^2}}dx = \frac{x}{2}\sqrt{x^2+a^2}-\frac{a^2}{2}\ln(x+\sqrt{x^2+a^2})+C \end{equation} 35.∫x2+a2x2dx=2xx2+a2−2a2ln(x+x2+a2)+C
36. ∫ x 2 ( x 2 + a 2 ) 3 d x = − x x 2 + a 2 + ln ( x + x 2 + a 2 ) + C \begin{equation} 36.\,\int\!\! \frac{x^2}{\sqrt{(x^2+a^2)^3}}dx= -\frac{x}{\sqrt{x^2+a^2}}+\ln (x+\sqrt{x^2+a^2})+C \end{equation} 36.∫(x2+a2)3x2dx=−x2+a2x+ln(x+x2+a2)+C
37. ∫ d x x x 2 + a 2 = 1 a ln x 2 + a 2 − a ∣ x ∣ + C \begin{equation} 37.\,\int\!\! \frac{dx}{x\sqrt{x^2+a^2}}= \frac{1}{a}\ln \frac{\sqrt{x^2+a^2}-a}{\vert x \vert} +C \end{equation} 37.∫xx2+a2dx=a1ln∣x∣x2+a2−a+C
38. ∫ d x x 2 x 2 + a 2 = − x 2 + a 2 a 2 x + C \begin{equation} 38.\,\int\!\! \frac{dx}{x^2\sqrt{x^2+a^2}}= -\frac{\sqrt{x^2+a^2}}{a^2x}+C \end{equation} 38.∫x2x2+a2dx=−a2xx2+a2+C
39. ∫ x 2 + a 2 d x = x 2 x 2 + a 2 + a 2 2 ln ( x + x 2 + a 2 ) + C \begin{equation} 39.\,\int\!\! \sqrt{x^2+a^2}dx=\frac{x}{2}\sqrt{x^2+a^2}+\frac{a^2}{2}\ln(x+\sqrt{x^2+a^2})+C \end{equation} 39.∫x2+a2dx=2xx2+a2+2a2ln(x+x2+a2)+C
40. ∫ ( x 2 + a 2 ) 3 d x = x 8 ( 2 x 2 + 5 a 2 ) x 2 + a 2 + 3 8 a 4 ln ( x + x 2 + a 2 ) + C \begin{equation} 40.\,\int\!\! \sqrt{(x^2+a^2)^3}dx = \frac{x}{8}(2x^2+5a^2)\sqrt{x^2+a^2}+ \frac{3}{8}a^4\ln(x+\sqrt{x^2+a^2})+C \end{equation} 40.∫(x2+a2)3dx=8x(2x2+5a2)x2+a2+83a4ln(x+x2+a2)+C
41. ∫ x x 2 + a 2 d x = 1 3 ( x 2 + a 2 ) 3 + C \begin{equation} 41.\,\int\!\! x\sqrt{x^2+a^2}dx= \frac{1}{3}\sqrt{(x^2+a^2)^3}+C \end{equation} 41.∫xx2+a2dx=31(x2+a2)3+C
42. ∫ x 2 x 2 + a 2 d x = x 8 ( 2 x 2 + a 2 ) x 2 + a 2 − a 4 8 ln ( x + x 2 + a 2 ) + C \begin{equation} 42.\,\int\!\! x^2 \sqrt{x^2+a^2}dx=\frac{x}{8}(2x^2+a^2)\sqrt{x^2+a^2}-\frac{a^4}{8}\ln(x+\sqrt{x^2+a^2})+C \end{equation} 42.∫x2x2+a2dx=8x(2x2+a2)x2+a2−8a4ln(x+x2+a2)+C
43. ∫ x 2 + a 2 x d x = x 2 + a 2 + a ln x 2 + a 2 − a ∣ x ∣ + C \begin{equation} 43.\,\int\!\! \frac{\sqrt{x^2+a^2}}{x}dx= \sqrt{x^2+a^2}+a\ln\frac{\sqrt{x^2+a^2}-a}{\vert x \vert} +C \end{equation} 43.∫xx2+a2dx=x2+a2+aln∣x∣x2+a2−a+C
44. ∫ x 2 + a 2 x 2 d x = − x 2 + a 2 x + ln ( x + x 2 + a 2 ) + C \begin{equation} 44.\,\int\!\! \frac{\sqrt{x^2+a^2}}{x^2}dx= -\frac{\sqrt{x^2+a^2}}{x} + \ln(x+\sqrt{x^2+a^2}) +C \end{equation} 44.∫x2x2+a2dx=−xx2+a2+ln(x+x2+a2)+C
含有 x 2 − a 2 ( a > 0 ) \sqrt{x^2-a^2} \quad (a>0) x2−a2(a>0) 的积分
45. ∫ d x x 2 − a 2 = x ∣ x ∣ a r c h ∣ x ∣ a + C 1 = l n ∣ x + x 2 − a 2 ∣ + C \begin{equation} 45.\,\int\!\! \frac{dx}{\sqrt{x^2-a^2}}= \frac{x}{\vert x \vert} arch \frac{\vert x \vert}{a}+C_1= ln \vert x+ \sqrt{x^2-a^2} \vert +C \end{equation} 45.∫x2−a2dx=∣x∣xarcha∣x∣+C1=ln∣x+x2−a2∣+C
46. ∫ d x ( x 2 − a 2 ) 3 = − x a 2 x 2 − a 2 + C \begin{equation} 46.\,\int\!\! \frac{dx}{\sqrt{(x^2-a^2)^3}}= - \frac{x}{a^2\sqrt{x^2-a^2}} +C \end{equation} 46.∫(x2−a2)3dx=−a2x2−a2x+C
47. ∫ x x 2 − a 2 d x = x 2 − a 2 + C \begin{equation} 47.\,\int\!\! \frac{x}{\sqrt{x^2-a^2}}dx = \sqrt{x^2-a^2}+C \end{equation} 47.∫x2−a2xdx=x2−a2+C
48. ∫ x ( x 2 − a 2 ) 3 d x = − 1 x 2 − a 2 + C \begin{equation} 48.\,\int\!\! \frac{x}{\sqrt{(x^2-a^2)^3}}dx = - \frac{1}{\sqrt{x^2-a^2}} +C \end{equation} 48.∫(x2−a2)3xdx=−x2−a21+C
49. ∫ x 2 x 2 − a 2 d x = x 2 x 2 − a 2 + a 2 2 ln ∣ x + x 2 − a 2 ∣ + C \begin{equation} 49.\,\int\!\! \frac{x^2}{\sqrt{x^2-a^2}}dx =\frac{x}{2}\sqrt{x^2 -a ^2} + \frac{a^2}{2} \ln \vert x+ \sqrt{x^2-a^2} \vert +C \end{equation} 49.∫x2−a2x2dx=2xx2−a2+2a2ln∣x+x2−a2∣+C
50. ∫ x 2 ( x 2 − a 2 ) 3 d x = − x x 2 − a 2 + ln ∣ x + x 2 − a 2 ∣ + C \begin{equation} 50.\,\int\!\! \frac{x^2}{\sqrt{(x^2-a^2)^3}}dx =-\frac{x}{\sqrt{x^2 -a ^2}} + \ln \vert x+ \sqrt{x^2-a^2} \vert +C \end{equation} 50.∫(x2−a2)3x2dx=−x2−a2x+ln∣x+x2−a2∣+C
51. ∫ d x x x 2 − a 2 = 1 a arccos a ∣ x ∣ + C \begin{equation} 51.\,\int\!\! \frac{dx}{x\sqrt{x^2-a^2}}= \frac{1}{a} \arccos \frac{a}{\vert x \vert} +C \end{equation} 51.∫xx2−a2dx=a1arccos∣x∣a+C
52. ∫ d x x 2 x 2 − a 2 = x 2 − a 2 a 2 x + C \begin{equation} 52.\,\int\!\! \frac{dx}{x^2\sqrt{x^2-a^2}} =\frac{ \sqrt {x^2-a^2}}{a^2x}+C \end{equation} 52.∫x2x2−a2dx=a2xx2−a2+C
53. ∫ x 2 − a 2 d x = x 2 x 2 − a 2 − a 2 2 ln ∣ x + x 2 − a 2 ∣ + C \begin{equation} 53.\,\int\!\! \sqrt{x^2-a^2}dx = \frac{x}{2} \sqrt{x^2-a^2} - \frac{a^2}{2} \ln \vert x+ \sqrt{x^2-a^2} \vert +C \end{equation} 53.∫x2−a2dx=2xx2−a2−2a2ln∣x+x2−a2∣+C
54. ∫ ( x 2 − a 2 ) 3 d x = x 8 ( 2 x 2 − 5 a 2 ) x 2 − a 2 + 3 8 a 4 ln ∣ x + x 2 − a 2 ∣ + C \begin{equation} 54.\,\int\!\! \sqrt{(x^2-a^2)^3}dx =\frac{x}{8}(2x^2-5a^2)\sqrt{x^2-a^2} + \frac{3}{8} a^4 \ln \vert x+ \sqrt{x^2-a^2} \vert +C \end{equation} 54.∫(x2−a2)3dx=8x(2x2−5a2)x2−a2+83a4ln∣x+x2−a2∣+C
55. ∫ x x 2 − a 2 d x = 1 3 ( x 2 − a 2 ) 3 + C \begin{equation} 55.\,\int\!\! x \sqrt{x^2-a^2}dx= \frac{1}{3}\sqrt{(x^2-a^2)^3} +C \end{equation} 55.∫xx2−a2dx=31(x2−a2)3+C
56. ∫ x 2 x 2 − a 2 d x = x 8 ( 2 x 2 − a 2 ) x 2 − a 2 − a 4 8 ln ∣ x + x 2 − a 2 ∣ + C \begin{equation} 56.\,\int\!\! x^2 \sqrt{x^2-a^2} dx = \frac{x}{8} (2x^2-a^2)\sqrt{x^2-a^2}-\frac{a^4}{8}\ln \vert x+ \sqrt{x^2-a^2} \vert +C \end{equation} 56.∫x2x2−a2dx=8x(2x2−a2)x2−a2−8a4ln∣x+x2−a2∣+C
57. ∫ x 2 − a 2 x d x = x 2 − a 2 − arccos a ∣ x ∣ + C \begin{equation} 57.\,\int\!\! \frac{\sqrt{x^2-a^2}}{x} dx= \sqrt{x^2-a^2}- \arccos \frac{a}{\vert x \vert} +C \end{equation} 57.∫xx2−a2dx=x2−a2−arccos∣x∣a+C
58. ∫ x 2 − a 2 x 2 d x = − x 2 − a 2 x + ln ∣ x + x 2 − a 2 ∣ + C \begin{equation} 58.\,\int\!\! \frac{\sqrt{x^2-a^2}}{x^2} dx = -\frac{\sqrt{x^2-a^2}}{x} + \ln \vert x + \sqrt{x^2-a^2} \vert +C \end{equation} 58.∫x2x2−a2dx=−xx2−a2+ln∣x+x2−a2∣+C
含有 a 2 − x 2 ( a > 0 ) \sqrt{a^2-x^2} \quad (a>0) a2−x2(a>0) 的积分
59. ∫ d x a 2 − x 2 = arcsin x a + C \begin{equation} 59.\,\int\!\! \frac{dx}{\sqrt{a^2-x^2}} =\arcsin \frac{x}{a} +C \end{equation} 59.∫a2−x2dx=arcsinax+C
60. ∫ d x ( a 2 − x 2 ) 3 = x a 2 a 2 − x 2 + C \begin{equation} 60.\,\int\!\! \frac{dx}{\sqrt{(a^2-x^2)^3}}= \frac{x}{a^2\sqrt{a^2-x^2}}+C \end{equation} 60.∫(a2−x2)3dx=a2a2−x2x+C
61. ∫ x a 2 − x 2 d x = − a 2 − x 2 + C \begin{equation} 61.\,\int\!\! \frac{x}{\sqrt{a^2-x^2}} dx = - \sqrt{a^2-x^2}+C \end{equation} 61.∫a2−x2xdx=−a2−x2+C
62. ∫ x ( a 2 − x 2 ) 3 d x = − 1 a 2 − x 2 + C \begin{equation} 62.\,\int\!\! \frac{x}{\sqrt{(a^2-x^2)^3}} dx = - \frac{1}{\sqrt{a^2-x^2}} +C \end{equation} 62.∫(a2−x2)3xdx=−a2−x21+C
63. ∫ x 2 a 2 − x 2 d x = − x 2 a 2 − x 2 + a 2 2 arcsin x a + C \begin{equation} 63.\,\int\!\! \frac{x^2}{\sqrt{a^2-x^2}}dx = -\frac{x}{2}\sqrt{a^2-x^2} + \frac{a^2}{2}\arcsin \frac{x}{a}+C \end{equation} 63.∫a2−x2x2dx=−2xa2−x2+2a2arcsinax+C
64. ∫ x 2 ( a 2 − x 2 ) 3 d x = x a 2 − x 2 − arcsin x a + C \begin{equation} 64.\,\int\!\! \frac{x^2}{\sqrt{(a^2-x^2)^3}}dx =\frac{x}{\sqrt{a^2-x^2}}- \arcsin \frac{x}{a} +C \end{equation} 64.∫(a2−x2)3x2dx=a2−x2x−arcsinax+C
65. ∫ d x x a 2 − x 2 = 1 a ln a − a 2 − x 2 ∣ x ∣ + C \begin{equation} 65.\,\int\!\! \frac{dx}{x\sqrt{a^2-x^2}}=\frac{1}{a}\ln\frac{a-\sqrt{a^2-x^2}}{\vert x \vert } +C \end{equation} 65.∫xa2−x2dx=a1ln∣x∣a−a2−x2+C
66. ∫ d x x 2 a 2 − x 2 = − a 2 − x 2 a 2 x + C \begin{equation} 66.\,\int\!\! \frac{dx}{x^2\sqrt{a^2-x^2}}=- \frac{a^2-x^2}{a^2x}+C \end{equation} 66.∫x2a2−x2dx=−a2xa2−x2+C
67. ∫ a 2 − x 2 d x = x 2 a 2 − x 2 + a 2 2 arcsin x a + C \begin{equation} 67.\,\int\!\! \sqrt{a^2-x^2}dx= \frac{x}{2} \sqrt{a^2-x^2} + \frac{a^2}{2}\arcsin \frac{x}{a}+C \end{equation} 67.∫a2−x2dx=2xa2−x2+2a2arcsinax+C
68. ∫ ( a 2 − x 2 ) 3 d x = x 8 ( 5 a 2 − 2 x 2 ) a 2 − x 2 + 3 8 a 4 arcsin x a + C \begin{equation} 68.\,\int\!\! \sqrt{(a^2-x^2)^3}dx=\frac{x}{8}(5a^2-2x^2)\sqrt{a^2-x^2}+\frac{3}{8}a^4\arcsin\frac{x}{a}+C \end{equation} 68.∫(a2−x2)3dx=8x(5a2−2x2)a2−x2+83a4arcsinax+C
69. ∫ x a 2 − x 2 d x = − 1 3 ( a 2 − x 2 ) 3 + C \begin{equation} 69.\,\int\!\! x\sqrt{a^2-x^2}dx=-\frac{1}{3}\sqrt{(a^2-x^2)^3}+C \end{equation} 69.∫xa2−x2dx=−31(a2−x2)3+C
70. ∫ x 2 a 2 − x 2 d x = x 8 ( 2 x 2 − a 2 ) a 2 − x 2 + a 4 8 arcsin x a + C \begin{equation} 70.\,\int\!\! x^2\sqrt{a^2-x^2}dx=\frac{x}{8}(2x^2-a^2)\sqrt{a^2-x^2}+\frac{a^4}{8}\arcsin\frac{x}{a}+C \end{equation} 70.∫x2a2−x2dx=8x(2x2−a2)a2−x2+8a4arcsinax+C
71. ∫ a 2 − x 2 x d x = a 2 − x 2 + a ln a − a 2 − x 2 ∣ x ∣ + C \begin{equation} 71.\,\int\!\! \frac{\sqrt{a^2-x^2}}{x} dx =\sqrt{a^2-x^2}+a \ln \frac{a-\sqrt{a^2-x^2}}{\vert x \vert}+C \end{equation} 71.∫xa2−x2dx=a2−x2+aln∣x∣a−a2−x2+C
72. ∫ a 2 − x 2 x 2 d x = − a 2 − x 2 x − arcsin x a + C \begin{equation} 72.\,\int\!\! \frac{\sqrt{a^2-x^2}}{x^2}dx=-\frac{\sqrt{a^2-x^2}}{x}-\arcsin \frac{x}{a}+C \end{equation} 72.∫x2a2−x2dx=−xa2−x2−arcsinax+C
含有 ± a x 2 + b x + c ( a > 0 ) \sqrt{\pm ax^2+bx+c} \quad (a>0) ±ax2+bx+c(a>0)的积分
73. ∫ d x a x 2 + b x + c = 1 a ln ∣ 2 a x + b + 2 a a x 2 + b x + c ∣ + C \begin{equation} 73.\,\int\!\! \frac{dx}{\sqrt{ax^2+bx+c}}=\frac{1}{\sqrt{a}}\ln \vert 2ax+b+2\sqrt{a}\sqrt{ax^2+bx+c}\,\vert +C \end{equation} 73.∫ax2+bx+cdx=a1ln∣2ax+b+2aax2+bx+c∣+C
74. ∫ a x 2 + b x + c d x = 2 a x + b 4 a a x 2 + b x + c + 4 a c − b 2 8 a 3 ln ∣ 2 a x + b + 2 a a x 2 + b x + c ∣ + C \begin{equation} 74.\,\int\!\! \sqrt{ax^2+bx+c}\,dx=\frac{2ax+b}{4a}\sqrt{ax^2+bx+c} +\frac{4ac-b^2}{8\sqrt{a^3}}\ln\vert 2ax+b+2\sqrt{a}\sqrt{ax^2+bx+c}\vert +C \end{equation} 74.∫ax2+bx+cdx=4a2ax+bax2+bx+c+8a34ac−b2ln∣2ax+b+2aax2+bx+c∣+C
75. ∫ x a x 2 + b x + c d x = 1 a a x 2 + b x + c − b 2 a 3 ln ∣ 2 a x + b + 2 a a x 2 + b x + c ∣ + C \begin{equation} 75.\,\int\!\! \frac{x}{\sqrt{ax^2 + bx + c}}dx = \frac{1}{a}\sqrt{ ax^2 + bx + c} -\frac{b}{2\sqrt{a^3}}\ln \vert 2ax + b + 2 \sqrt{a}\sqrt{ax^2 + bx + c}\,\vert +C \end{equation} 75.∫ax2+bx+cxdx=a1ax2+bx+c−2a3bln∣2ax+b+2aax2+bx+c∣+C
76. ∫ d x c + b x − a x 2 = 1 a arcsin 2 a x − b b 2 + 4 a c + C \begin{equation} 76.\,\int\!\! \frac{dx}{\sqrt{c+bx-ax^2}}= \frac{1}{\sqrt{a}}\arcsin\frac{2ax-b}{\sqrt{b^2+4ac}}+C \end{equation} 76.∫c+bx−ax2dx=a1arcsinb2+4ac2ax−b+C
77. ∫ c + b x − a x 2 d x = 2 a x − b 4 a c + b x − a x 2 + b 2 + 4 a c 8 a 3 arcsin 2 a x − b b 2 + 4 a c + C \begin{equation} 77.\,\int\!\! \sqrt{c+bx-ax^2}\,dx=\frac{2ax-b}{4a}\sqrt{c+bx-ax^2}+\frac{b^2+4ac}{8\sqrt{a^3}}\arcsin \frac{2ax-b}{\sqrt{b^2+4ac}}+C \end{equation} 77.∫c+bx−ax2dx=4a2ax−bc+bx−ax2+8a3b2+4acarcsinb2+4ac2ax−b+C
78. ∫ x c + b x − a x 2 d x = − 1 a c + b x − a x 2 + b 2 a 3 arcsin 2 a x − b b 2 + 4 a c + C \begin{equation} 78.\,\int\!\! \frac{x}{\sqrt{c+bx-ax^2}}\,dx=-\frac{1}{a}\sqrt{c+bx-ax^2}+\frac{b}{2\sqrt{a^3}}\arcsin\frac{2ax-b}{\sqrt{b^2+4ac}}+C \end{equation} 78.∫c+bx−ax2xdx=−a1c+bx−ax2+2a3barcsinb2+4ac2ax−b+C
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