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积分表

(一) 含有 ax+bax+b 的积分​

  1. ∫dxax+b=1aln⁡∣ax+b∣+C\int \frac{\mathrm{d} x}{ax+b} = \frac{1}{a} \ln |ax+b| + C

  2. ∫(ax+b)μdx=1a(μ+1)(ax+b)μ+1+C(μ≠−1)\int (ax+b)^{\mu} \mathrm{d} x = \frac{1}{a(\mu+1)} (ax+b)^{\mu+1} + C \quad (\mu \neq -1)

  3. ∫xax+bdx=1a2(ax+b−bln⁡∣ax+b∣)+C\int \frac{x}{ax+b} \mathrm{d} x = \frac{1}{a^2} \left(ax + b - b \ln |ax+b| \right) + C

  4. ∫x2ax+bdx=1a3[12(ax+b)2−2b(ax+b)+b2ln⁡∣ax+b∣]+C\int \frac{x^2}{ax+b} \mathrm{d} x = \frac{1}{a^3} \left[ \frac{1}{2} (ax+b)^2 - 2b (ax+b) + b^2 \ln |ax+b| \right] + C

  5. ∫dxx(ax+b)=−1bln⁡∣ax+bx∣+C\int \frac{\mathrm{d} x}{x(ax+b)} = -\frac{1}{b} \ln \left| \frac{ax+b}{x} \right| + C

  6. ∫dxx2(ax+b)=−1bx+ab2ln⁡∣ax+bx∣+C\int \frac{\mathrm{d} x}{x^2(ax+b)} = -\frac{1}{bx} + \frac{a}{b^2} \ln \left| \frac{ax+b}{x} \right| + C

  7. ∫x(ax+b)2dx=1a2(ln⁡∣ax+b∣+bax+b)+C\int \frac{x}{(ax+b)^2} \mathrm{d} x = \frac{1}{a^2} \left( \ln |ax+b| + \frac{b}{ax+b} \right) + C

  8. ∫x2(ax+b)2dx=1a3(ax+b−2bln⁡∣ax+b∣−b2ax+b)+C\int \frac{x^2}{(ax+b)^2} \mathrm{d} x = \frac{1}{a^3} \left(ax + b - 2b \ln |ax+b| - \frac{b^2}{ax+b} \right) + C

  9. ∫dxx(ax+b)2=1b(ax+b)−1b2ln⁡∣ax+bx∣+C\int \frac{\mathrm{d} x}{x(ax+b)^2} = \frac{1}{b(ax+b)} - \frac{1}{b^2} \ln \left| \frac{ax+b}{x} \right| + C

(二) 含有 ax+b\sqrt{ax+b} 的积分​

  1. ∫ax+b dx=23a(ax+b)3+C\int \sqrt{ax+b} \, \mathrm{d} x = \frac{2}{3a} \sqrt{(ax+b)^3} + C

  2. ∫xax+b dx=215a2(3ax−2b)(ax+b)3+C\int x \sqrt{ax+b} \, \mathrm{d} x = \frac{2}{15a^2} (3ax - 2b) \sqrt{(ax+b)^3} + C

  3. ∫x2ax+b dx=2105a3(15a2x2−12abx+8b2)(ax+b)3+C\int x^2 \sqrt{ax+b} \, \mathrm{d} x = \frac{2}{105a^3} (15a^2 x^2 - 12abx + 8b^2) \sqrt{(ax+b)^3} + C

  4. ∫xax+b dx=23a2(ax−2b)ax+b+C\int \frac{x}{\sqrt{ax+b}} \, \mathrm{d} x = \frac{2}{3a^2} (ax - 2b) \sqrt{ax+b} + C

  5. ∫x2ax+b dx=215a3(3a2x2−4abx+8b2)ax+b+C\int \frac{x^2}{\sqrt{ax+b}} \, \mathrm{d} x = \frac{2}{15a^3} (3a^2 x^2 - 4abx + 8b^2) \sqrt{ax+b} + C

  6. ∫dxxax+b={1bln⁡∣ax+b−bax+b+b∣+C(b>0)2−barctan⁡(ax+b−b)+C(b<0)\int \frac{\mathrm{d} x}{x \sqrt{ax+b}} = \begin{cases} \frac{1}{\sqrt{b}} \ln \left| \frac{\sqrt{ax+b} - \sqrt{b}}{\sqrt{ax+b} + \sqrt{b}} \right| + C & (b > 0) \\ \frac{2}{\sqrt{-b}} \arctan \left( \sqrt{\frac{ax+b}{-b}} \right) + C & (b < 0) \end{cases}

  7. ∫dxx2ax+b=−ax+bbx−a2b∫dxxax+b\int \frac{\mathrm{d} x}{x^2 \sqrt{ax+b}} = -\frac{\sqrt{ax+b}}{bx} - \frac{a}{2b} \int \frac{\mathrm{d} x}{x \sqrt{ax+b}}

  8. ∫ax+bx dx=2ax+b+b∫dxxax+b\int \frac{\sqrt{ax+b}}{x} \, \mathrm{d} x = 2 \sqrt{ax+b} + b \int \frac{\mathrm{d} x}{x \sqrt{ax+b}}

  9. ∫ax+bx2 dx=−ax+bx+a2∫dxxax+b\int \frac{\sqrt{ax+b}}{x^2} \, \mathrm{d} x = -\frac{\sqrt{ax+b}}{x} + \frac{a}{2} \int \frac{\mathrm{d} x}{x \sqrt{ax+b}}

(三) 含有 x2±a2x^2 \pm a^2 的积分​

  1. ∫dxx2+a2=1aarctan⁡xa+C\int \frac{\mathrm{d} x}{x^2 + a^2} = \frac{1}{a} \arctan \frac{x}{a} + C

  2. ∫dx(x2+a2)n=x2(n−1)a2(x2+a2)n−1+2n−32(n−1)a2∫dx(x2+a2)n−1\int \frac{\mathrm{d} x}{(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{\mathrm{d} x}{(x^2 + a^2)^{n-1}}

  3. ∫dxx2−a2=12aln⁡∣x−ax+a∣+C\int \frac{\mathrm{d} x}{x^2 - a^2} = \frac{1}{2a} \ln \left| \frac{x-a}{x+a} \right| + C

(四) 含有 ax2+b(a>0)ax^2 + b \quad (a > 0) 的积分​

  1. ∫dxax2+b={1abarctan⁡abx+C(b>0)12−abln⁡∣ax−−bax+−b∣+C(b<0)\int \frac{\mathrm{d} x}{ax^2 + b} = \begin{cases} \frac{1}{\sqrt{ab}} \arctan \sqrt{\frac{a}{b}} x + C & (b > 0) \\ \frac{1}{2 \sqrt{-ab}} \ln \left| \frac{\sqrt{a} x - \sqrt{-b}}{\sqrt{a} x + \sqrt{-b}} \right| + C & (b < 0) \end{cases}

  2. ∫xax2+bdx=12aln⁡∣ax2+b∣+C\int \frac{x}{ax^2 + b} \mathrm{d} x = \frac{1}{2a} \ln \left| ax^2 + b \right| + C

  3. ∫x2ax2+bdx=xa−ba∫dxax2+b\int \frac{x^2}{ax^2 + b} \mathrm{d} x = \frac{x}{a} - \frac{b}{a} \int \frac{\mathrm{d} x}{ax^2 + b}

  4. ∫dxx(ax2+b)=12bln⁡x2∣ax2+b∣+C\int \frac{\mathrm{d} x}{x(ax^2 + b)} = \frac{1}{2b} \ln \frac{x^2}{| ax^2 + b |} + C

  5. ∫dxx2(ax2+b)=−1bx−ab∫dxax2+b\int \frac{\mathrm{d} x}{x^2 (ax^2 + b)} = -\frac{1}{bx} - \frac{a}{b} \int \frac{\mathrm{d} x}{ax^2 + b}

  6. ∫dxx3(ax2+b)=a2b2ln⁡∣ax2+b∣x2−12bx2+C\int \frac{\mathrm{d} x}{x^3 (ax^2 + b)} = \frac{a}{2b^2} \ln \frac{| ax^2 + b |}{x^2} - \frac{1}{2bx^2} + C

  7. ∫dx(ax2+b)2=x2b(ax2+b)+12b∫dxax2+b\int \frac{\mathrm{d} x}{(ax^2 + b)^2} = \frac{x}{2b(ax^2 + b)} + \frac{1}{2b} \int \frac{\mathrm{d} x}{ax^2 + b}

(五) 含有 ax2+bx+c(a>0)ax^2 + bx + c \quad (a > 0) 的积分​

  1. ∫dxax2+bx+c={24ac−b2arctan⁡(2ax+b4ac−b2)+C(b2<4ac)1b2−4acln⁡∣2ax+b−b2−4ac2ax+b+b2−4ac∣+C(b2>4ac)\int \frac{\mathrm{d} x}{ax^2 + bx + c} = \begin{cases} \frac{2}{\sqrt{4ac - b^2}} \arctan \left( \frac{2ax + b}{\sqrt{4ac - b^2}} \right) + C & (b^2 < 4ac) \\ \frac{1}{\sqrt{b^2 - 4ac}} \ln \left| \frac{2ax + b - \sqrt{b^2 - 4ac}}{2ax + b + \sqrt{b^2 - 4ac}} \right| + C & (b^2 > 4ac) \end{cases}

  2. ∫xax2+bx+cdx=12aln⁡∣ax2+bx+c∣−b2a∫dxax2+bx+c\int \frac{x}{ax^2 + bx + c} \mathrm{d} x = \frac{1}{2a} \ln \left| ax^2 + bx + c \right| - \frac{b}{2a} \int \frac{\mathrm{d} x}{ax^2 + bx + c}

(六) 含有 x2+a2(a>0)\sqrt{x^2+a^2} \quad (a>0) 的积分​

  1. ∫dxx2+a2=arshxa+C1=ln⁡(x+x2+a2)+C\int \frac{\mathrm{d} x}{\sqrt{x^2+a^2}} = \text{arsh} \frac{x}{a} + C_1 = \ln(x + \sqrt{x^2 + a^2}) + C

  2. ∫dx(x2+a2)3=xa2x2+a2+C\int \frac{\mathrm{d} x}{\sqrt{(x^2+a^2)^3}} = \frac{x}{a^2 \sqrt{x^2 + a^2}} + C

  3. ∫xx2+a2 dx=x2+a2+C\int \frac{x}{\sqrt{x^2 + a^2}} \, \mathrm{d} x = \sqrt{x^2 + a^2} + C

  4. ∫x(x2+a2)3dx=−1x2+a2+C\int \frac{x}{\sqrt{(x^2 + a^2)^3}} \mathrm{d} x = -\frac{1}{\sqrt{x^2 + a^2}} + C

  5. ∫x2x2+a2 dx=x2x2+a2−a22ln⁡(x+x2+a2)+C\int \frac{x^2}{\sqrt{x^2 + a^2}} \, \mathrm{d} x = \frac{x}{2} \sqrt{x^2 + a^2} - \frac{a^2}{2} \ln(x + \sqrt{x^2 + a^2}) + C

  6. ∫x2(x2+a2)3 dx=−xx2+a2+ln⁡(x+x2+a2)+C\int \frac{x^2}{\sqrt{(x^2 + a^2)^3}} \, \mathrm{d} x = -\frac{x}{\sqrt{x^2 + a^2}} + \ln(x + \sqrt{x^2 + a^2}) + C

  7. ∫dxxx2+a2=1aln⁡x2+a2−a∣x∣+C\int \frac{\mathrm{d} x}{x \sqrt{x^2 + a^2}} = \frac{1}{a} \ln \frac{\sqrt{x^2 + a^2} - a}{|x|} + C

  8. ∫dxx2x2+a2=−x2+a2a2x+C\int \frac{\mathrm{d} x}{x^2 \sqrt{x^2 + a^2}} = -\frac{\sqrt{x^2 + a^2}}{a^2 x} + C

  9. ∫x2+a2 dx=x2x2+a2+a22ln⁡(x+x2+a2)+C\int \sqrt{x^2 + a^2} \, \mathrm{d} x = \frac{x}{2} \sqrt{x^2 + a^2} + \frac{a^2}{2} \ln(x + \sqrt{x^2 + a^2}) + C

  10. ∫(x2+a2)3 dx=x8(2x2+5a2)x2+a2+38a4ln⁡(x+x2+a2)+C\int \sqrt{(x^2 + a^2)^3} \, \mathrm{d} x = \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

  11. ∫xx2+a2 dx=13(x2+a2)3+C\int x \sqrt{x^2 + a^2} \, \mathrm{d} x = \frac{1}{3} \sqrt{(x^2 + a^2)^3} + C

  12. ∫x2x2+a2 dx=x8(2x2+a2)x2+a2−a48ln⁡(x+x2+a2)+C\int x^2 \sqrt{x^2 + a^2} \, \mathrm{d} x = \frac{x}{8} (2x^2 + a^2) \sqrt{x^2 + a^2} - \frac{a^4}{8} \ln(x + \sqrt{x^2 + a^2}) + C

  13. ∫x2+a2x dx=x2+a2+aln⁡x2+a2−a∣x∣+C\int \frac{\sqrt{x^2 + a^2}}{x} \, \mathrm{d} x = \sqrt{x^2 + a^2} + a \ln \frac{\sqrt{x^2 + a^2} - a}{|x|} + C

  14. ∫x2+a2x2 dx=−x2+a2x+ln⁡(x+x2+a2)+C\int \frac{\sqrt{x^2 + a^2}}{x^2} \, \mathrm{d} x = -\frac{\sqrt{x^2 + a^2}}{x} + \ln(x + \sqrt{x^2 + a^2}) + C

(七) 含有 x2−a2(a>0)\sqrt{x^2 - a^2} \quad (a > 0) 的积分​

  1. ∫dxx2−a2=x∣x∣arch∣x∣a+C1=ln⁡∣x+x2−a2∣+C\int \frac{\mathrm{d} x}{\sqrt{x^2 - a^2}} = \frac{x}{|x|} \text{arch} \frac{|x|}{a} + C_1 = \ln \left| x + \sqrt{x^2 - a^2} \right| + C

  2. ∫dx(x2−a2)3=−xa2x2−a2+C\int \frac{\mathrm{d} x}{\left( \sqrt{x^2 - a^2} \right)^3} = - \frac{x}{a^2 \sqrt{x^2 - a^2}} + C

  3. ∫xx2−a2dx=x2−a2+C\int \frac{x}{\sqrt{x^2 - a^2}} \mathrm{d} x = \sqrt{x^2 - a^2} + C

  4. ∫x(x2−a2)3dx=−1x2−a2+C\int \frac{x}{\sqrt{(x^2 - a^2)^3}} \mathrm{d} x = - \frac{1}{\sqrt{x^2 - a^2}} + C

  5. ∫x2x2−a2dx=x2x2−a2+a22ln⁡∣x+x2−a2∣+C\int \frac{x^2}{\sqrt{x^2 - a^2}} \mathrm{d} x = \frac{x}{2} \sqrt{x^2 - a^2} + \frac{a^2}{2} \ln \left| x + \sqrt{x^2 - a^2} \right| + C

  6. ∫x2(x2−a2)3dx=−xx2−a2+ln⁡∣x+x2−a2∣+C\int \frac{x^2}{\left( \sqrt{x^2 - a^2} \right)^3} \mathrm{d} x = - \frac{x}{\sqrt{x^2 - a^2}} + \ln \left| x + \sqrt{x^2 - a^2} \right| + C

  7. ∫dxxx2−a2=1aarccosa∣x∣+C\int \frac{\mathrm{d} x}{x \sqrt{x^2 - a^2}} = \frac{1}{a} \text{arccos} \frac{a}{|x|} + C

  8. ∫dxx2x2−a2=x2−a2a2x+C\int \frac{\mathrm{d} x}{x^2 \sqrt{x^2 - a^2}} = \frac{\sqrt{x^2 - a^2}}{a^2 x} + C

  9. ∫x2−a2dx=x2x2−a2−a22ln⁡∣x+x2−a2∣+C\int \sqrt{x^2 - a^2} \mathrm{d} x = \frac{x}{2} \sqrt{x^2 - a^2} - \frac{a^2}{2} \ln \left| x + \sqrt{x^2 - a^2} \right| + C

  10. ∫(x2−a2)3 dx=x8(2x2−5a2)x2−a2+38a4ln⁡∣x+x2−a2∣+C\int \sqrt{(x^2 - a^2)^3} \, \mathrm{d} x = \frac{x}{8} \left(2x^2 - 5a^2\right) \sqrt{x^2 - a^2} + \frac{3}{8} a^4 \ln \left| x + \sqrt{x^2 - a^2} \right| + C

  11. ∫xx2−a2 dx=13(x2−a2)3+C\int x \sqrt{x^2 - a^2} \, \mathrm{d} x = \frac{1}{3} \sqrt{(x^2 - a^2)^3} + C

  12. ∫x2x2−a2 dx=x8(2x2−a2)x2−a2−a48ln⁡∣x+x2−a2∣+C\int x^2 \sqrt{x^2 - a^2} \, \mathrm{d} x = \frac{x}{8} \left(2x^2 - a^2\right) \sqrt{x^2 - a^2} - \frac{a^4}{8} \ln \left| x + \sqrt{x^2 - a^2} \right| + C

  13. ∫x2−a2x dx=x2−a2−aarccos⁡a∣x∣+C\int \frac{\sqrt{x^2 - a^2}}{x} \, \mathrm{d} x = \sqrt{x^2 - a^2} - a \arccos \frac{a}{|x|} + C

  14. ∫x2−a2x2 dx=−x2−a2x+ln⁡∣x+x2−a2∣+C\int \frac{\sqrt{x^2 - a^2}}{x^2} \, \mathrm{d} x = -\frac{\sqrt{x^2 - a^2}}{x} + \ln \left| x + \sqrt{x^2 - a^2} \right| + C

(八) 含有 a2−x2(a>0)\sqrt{a^2 - x^2} \quad (a > 0) 的积分​

  1. ∫dxa2−x2=arcsin⁡xa+C\int \frac{\mathrm{d} x}{\sqrt{a^2 - x^2}} = \arcsin \frac{x}{a} + C

  2. ∫dx(a2−x2)3=xa2a2−x2+C\int \frac{\mathrm{d} x}{\left( \sqrt{a^2 - x^2} \right)^3} = \frac{x}{a^2 \sqrt{a^2 - x^2}} + C

  3. ∫xa2−x2dx=−a2−x2+C\int \frac{x}{\sqrt{a^2 - x^2}} \mathrm{d} x = -\sqrt{a^2 - x^2} + C

  4. ∫x(a2−x2)3dx=1a2−x2+C\int \frac{x}{\sqrt{(a^2 - x^2)^3}} \mathrm{d} x = \frac{1}{\sqrt{a^2 - x^2}} + C

  5. ∫x2a2−x2dx=−x2a2−x2+a22arcsin⁡xa+C\int \frac{x^2}{\sqrt{a^2 - x^2}} \mathrm{d} x = -\frac{x}{2} \sqrt{a^2 - x^2} + \frac{a^2}{2} \arcsin \frac{x}{a} + C

  6. ∫x2(a2−x2)3dx=xa2−x2−arcsin⁡xa+C\int \frac{x^2}{\sqrt{(a^2 - x^2)^3}} \mathrm{d} x = \frac{x}{\sqrt{a^2 - x^2}} - \arcsin \frac{x}{a} + C

  7. ∫dxxa2−x2=1aln⁡a−a2−x2∣x∣+C\int \frac{\mathrm{d} x}{x \sqrt{a^2 - x^2}} = \frac{1}{a} \ln \frac{a - \sqrt{a^2 - x^2}}{|x|} + C

  8. ∫dxx2a2−x2=−a2−x2a2x+C\int \frac{\mathrm{d} x}{x^2 \sqrt{a^2 - x^2}} = -\frac{\sqrt{a^2 - x^2}}{a^2 x} + C

  9. ∫a2−x2dx=x2a2−x2+a22arcsin⁡xa+C\int \sqrt{a^2 - x^2} \mathrm{d} x = \frac{x}{2} \sqrt{a^2 - x^2} + \frac{a^2}{2} \arcsin \frac{x}{a} + C

  10. ∫(a2−x2)3dx=x8(5a2−2x2)a2−x2+38a4arcsin⁡xa+C\int \sqrt{\left( a^2 - x^2 \right)^3} \mathrm{d} x = \frac{x}{8} \left( 5a^2 - 2x^2 \right) \sqrt{a^2 - x^2} + \frac{3}{8} a^4 \arcsin \frac{x}{a} + C

  11. ∫xa2−x2dx=−13(a2−x2)3+C\int x \sqrt{a^2 - x^2} \mathrm{d} x = -\frac{1}{3} \sqrt{\left( a^2 - x^2 \right)^3} + C

  12. ∫x2a2−x2dx=x8(2x2−a2)a2−x2+a48arcsin⁡xa+C\int x^2 \sqrt{a^2 - x^2} \mathrm{d} x = \frac{x}{8} (2x^2 - a^2) \sqrt{a^2 - x^2} + \frac{a^4}{8} \arcsin \frac{x}{a} + C

  13. ∫a2−x2xdx=a2−x2+aln⁡a−a2−x2∣x∣+C\int \frac{\sqrt{a^2 - x^2}}{x} \mathrm{d} x = \sqrt{a^2 - x^2} + a \ln \frac{a - \sqrt{a^2 - x^2}}{|x|} + C

  14. ∫a2−x2x2dx=−a2−x2x−arcsin⁡xa+C\int \frac{\sqrt{a^2 - x^2}}{x^2} \mathrm{d} x = - \frac{\sqrt{a^2 - x^2}}{x} - \arcsin \frac{x}{a} + C

(九) 含有 ±ax2+bx+c(a>0)\sqrt{\pm ax^2 + bx + c} \quad (a > 0) 的积分​

  1. ∫dxax2+bx+c=1aln⁡∣2ax+b+2aax2+bx+c∣+C\int \frac{\mathrm{d} x}{\sqrt{ax^2 + bx + c}} = \frac{1}{\sqrt{a}} \ln \left| 2ax + b + 2 \sqrt{a} \sqrt{ax^2 + bx + c} \right| + C

  2. ∫ax2+bx+cdx=2ax+b4aax2+bx+c+4ac−b28a3ln⁡∣2ax+b+2aax2+bx+c∣+C\int \sqrt{ax^2 + bx + c} \mathrm{d} x = \frac{2ax + b}{4a} \sqrt{ax^2 + bx + c} + \frac{4ac - b^2}{8 \sqrt{a^3}} \ln \left| 2ax + b + 2 \sqrt{a} \sqrt{ax^2 + bx + c} \right| + C

  3. ∫xax2+bx+cdx=1aax2+bx+c−b2a3ln⁡∣2ax+b+2aax2+bx+c∣+C\int \frac{x}{\sqrt{ax^2 + bx + c}} \mathrm{d} x = \frac{1}{a} \sqrt{ax^2 + bx + c} - \frac{b}{2 \sqrt{a^3}} \ln \left| 2ax + b + 2 \sqrt{a} \sqrt{ax^2 + bx + c} \right| + C

  4. ∫dxc+bx−ax2=1aarcsin⁡2ax−bb2+4ac+C\int \frac{\mathrm{d} x}{\sqrt{c + bx - ax^2}} = \frac{1}{\sqrt{a}} \arcsin \frac{2ax - b}{\sqrt{b^2 + 4ac}} + C

  5. ∫c+bx−ax2dx=2ax−b4ac+bx−ax2+b2+4ac8a3arcsin⁡2ax−bb2+4ac+C\int \sqrt{c + bx - ax^2} \mathrm{d} x = \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

  6. ∫xc+bx−ax2dx=−1ac+bx−ax2+b2a3arcsin⁡2ax−bb2+4ac+C\int \frac{x}{\sqrt{c + bx - ax^2}} \mathrm{d} x = - \frac{1}{a} \sqrt{c + bx - ax^2} + \frac{b}{2 \sqrt{a^3}} \arcsin \frac{2ax - b}{\sqrt{b^2 + 4ac}} + C

(十) 含有 ±x−ax−b\sqrt{\pm \frac{x-a}{x-b}} 或 (x−a)(b−x)\sqrt{(x-a)(b-x)} 的积分​

  1. ∫x−ax−b dx=(x−b)x−ax−b+(b−a)ln⁡(∣x−a∣+∣x−b∣)+C\int \sqrt{\frac{x-a}{x-b}} \, \mathrm{d} x = (x-b) \sqrt{\frac{x-a}{x-b}} + (b-a) \ln \left( \sqrt{|x-a|} + \sqrt{|x-b|} \right) + C

  2. ∫x−ab−x dx=(x−b)x−ab−x+(b−a)arcsin⁡x−ab−a+C\int \sqrt{\frac{x-a}{b-x}} \, \mathrm{d} x = (x-b) \sqrt{\frac{x-a}{b-x}} + (b-a) \arcsin \sqrt{\frac{x-a}{b-a}} + C

  3. ∫dx(x−a)(b−x)=2arcsin⁡x−ab−a+C(a<b)\int \frac{\mathrm{d} x}{\sqrt{(x-a)(b-x)}} = 2 \arcsin \sqrt{\frac{x-a}{b-a}} + C \quad (a < b)

  4. ∫(x−a)(b−x) dx=2x−a−b4(x−a)(b−x)+(b−a)24arcsin⁡x−ab−a+C(a<b)\int \sqrt{(x-a)(b-x)} \, \mathrm{d} x = \frac{2x-a-b}{4} \sqrt{(x-a)(b-x)} + \frac{(b-a)^2}{4} \arcsin \sqrt{\frac{x-a}{b-a}} + C \quad (a < b)

(十一) 含有三角函数的积分​

  1. ∫sin⁡x dx=−cos⁡x+C\int \sin x \, \mathrm{d} x = -\cos x + C

  2. ∫cos⁡x dx=sin⁡x+C\int \cos x \, \mathrm{d} x = \sin x + C

  3. ∫tan⁡x dx=−ln⁡∣cos⁡x∣+C\int \tan x \, \mathrm{d} x = -\ln | \cos x | + C

  4. ∫cot⁡x dx=ln⁡∣sin⁡x∣+C\int \cot x \, \mathrm{d} x = \ln | \sin x | + C

  5. ∫sec⁡x dx=ln⁡∣tan⁡(π4+x2)∣+C=ln⁡∣sec⁡x+tan⁡x∣+C\int \sec x \, \mathrm{d} x = \ln \left| \tan \left( \frac{\pi}{4} + \frac{x}{2} \right) \right| + C = \ln | \sec x + \tan x | + C

  6. ∫csc⁡x dx=ln⁡∣tan⁡x2∣+C=ln⁡∣csc⁡x−cot⁡x∣+C\int \csc x \, \mathrm{d} x = \ln \left| \tan \frac{x}{2} \right| + C = \ln | \csc x - \cot x | + C

  7. ∫sec⁡2x dx=tan⁡x+C\int \sec^2 x \, \mathrm{d} x = \tan x + C

  8. ∫csc⁡2x dx=−cot⁡x+C\int \csc^2 x \, \mathrm{d} x = -\cot x + C

  9. ∫sec⁡xtan⁡x dx=sec⁡x+C\int \sec x \tan x \, \mathrm{d} x = \sec x + C

  10. ∫csc⁡xcot⁡x dx=−csc⁡x+C\int \csc x \cot x \, \mathrm{d} x = -\csc x + C

  11. ∫sin⁡2x dx=x2−14sin⁡2x+C\int \sin^2 x \, \mathrm{d} x = \frac{x}{2} - \frac{1}{4} \sin 2x + C

  12. ∫cos⁡2x dx=x2+14sin⁡2x+C\int \cos^2 x \, \mathrm{d} x = \frac{x}{2} + \frac{1}{4} \sin 2x + C

  13. ∫sin⁡nx dx=1nsin⁡n−1xcos⁡x+n−1n∫sin⁡n−2x dx\int \sin^n x \, \mathrm{d} x = \frac{1}{n} \sin^{n-1} x \cos x + \frac{n-1}{n} \int \sin^{n-2} x \, \mathrm{d} x

  14. ∫cos⁡nx dx=1ncos⁡n−1xsin⁡x+n−1n∫cos⁡n−2x dx\int \cos^n x \, \mathrm{d} x = \frac{1}{n} \cos^{n-1} x \sin x + \frac{n-1}{n} \int \cos^{n-2} x \, \mathrm{d} x

  15. ∫dxsin⁡nx=−1n−1⋅cos⁡xsin⁡n−1x+n−2n−1∫dxsin⁡n−2x\int \frac{\mathrm{d} x}{\sin^n x} = -\frac{1}{n-1} \cdot \frac{\cos x}{\sin^{n-1} x} + \frac{n-2}{n-1} \int \frac{\mathrm{d} x}{\sin^{n-2} x}

  16. ∫dxcos⁡nx=1n−1⋅sin⁡xcos⁡n−1x+n−2n−1∫dxcos⁡n−2x\int \frac{\mathrm{d} x}{\cos^n x} = \frac{1}{n-1} \cdot \frac{\sin x}{\cos^{n-1} x} + \frac{n-2}{n-1} \int \frac{\mathrm{d} x}{\cos^{n-2} x}

  17. ∫cos⁡mxsin⁡nx dx=1m+ncos⁡m−1xsin⁡n+1x+m−1m+n∫cos⁡m−2xsin⁡nx dx=−1m+ncos⁡m+1xsin⁡n−1x+n−1m+n∫cos⁡mxsin⁡n−2x dx\int \cos^m x \sin^n x \, \mathrm{d} x \\ = \frac{1}{m+n} \cos^{m-1} x \sin^{n+1} x + \frac{m-1}{m+n} \int \cos^{m-2} x \sin^n x \, \mathrm{d} x \\ = -\frac{1}{m+n} \cos^{m+1} x \sin^{n-1} x + \frac{n-1}{m+n} \int \cos^m x \sin^{n-2} x \, \mathrm{d} x

  18. ∫sin⁡axcos⁡bx dx=−12(a+b)cos⁡(a+b)x−12(a−b)cos⁡(a−b)x+C\int \sin ax \cos bx \, \mathrm{d} x = -\frac{1}{2(a+b)} \cos (a+b)x - \frac{1}{2(a-b)} \cos (a-b)x + C

  19. ∫sin⁡axsin⁡bx dx=−12(a+b)sin⁡(a+b)x+12(a−b)sin⁡(a−b)x+C\int \sin ax \sin bx \, \mathrm{d} x = -\frac{1}{2(a+b)} \sin (a+b)x + \frac{1}{2(a-b)} \sin (a-b)x + C

  20. ∫cos⁡axcos⁡bxdx=12(a+b)sin⁡(a+b)x+12(a−b)sin⁡(a−b)x+C\int \cos ax \cos bx\mathrm{d} x = \frac{1}{2(a+b)}\sin (a+b)x + \frac{1}{2(a-b)}\sin (a-b)x + C

  21. ∫dxa+bsin⁡x=2a2−b2arctan⁡atan⁡x2+ba2−b2+C(a2>b2)\int \frac{\mathrm{d} x}{a+b\sin x} = \frac{2}{\sqrt{a^2-b^2}} \arctan \frac{a \tan \frac{x}{2} + b}{\sqrt{a^2-b^2}} + C \quad (a^2 > b^2)

  22. ∫dxa+bsin⁡x=1b2−a2ln⁡∣atan⁡x2+b−b2−a2atan⁡x2+b+b2−a2∣+C(a2<b2)\int \frac{\mathrm{d} x}{a+b\sin x} = \frac{1}{\sqrt{b^2-a^2}} \ln \left| \frac{a \tan \frac{x}{2} + b - \sqrt{b^2-a^2}}{a \tan \frac{x}{2} + b + \sqrt{b^2-a^2}} \right| + C \quad (a^2 < b^2)

  23. ∫dxa+bcos⁡x=2a+ba+ba−barctan⁡(a−ba+btan⁡x2)+C(a2>b2)\int \frac{\mathrm{d} x}{a+b\cos x} = \frac{2}{a+b} \sqrt{\frac{a+b}{a-b}} \arctan \left( \sqrt{\frac{a-b}{a+b}} \tan \frac{x}{2} \right) + C \quad (a^2 > b^2)

  24. ∫dxa+bcos⁡x=1a+ba+bb−aln⁡∣tan⁡x2+a+bb−atan⁡x2−a+bb−a∣+C(a2<b2)\int \frac{\mathrm{d} x}{a+b\cos x} = \frac{1}{a+b} \sqrt{\frac{a+b}{b-a}} \ln \left| \frac{\tan \frac{x}{2} + \sqrt{\frac{a+b}{b-a}}}{\tan \frac{x}{2} - \sqrt{\frac{a+b}{b-a}}} \right| + C \quad (a^2 < b^2)

  25. ∫dxa2cos⁡2x+b2sin⁡2x=1abarctan⁡(batan⁡x)+C\int \frac{\mathrm{d} x}{a^2 \cos^2 x + b^2 \sin^2 x} = \frac{1}{ab} \arctan \left( \frac{b}{a} \tan x \right) + C

  26. ∫dxa2cos⁡2x−b2sin⁡2x=12abln⁡∣btan⁡x+abtan⁡x−a∣+C\int \frac{\mathrm{d} x}{a^2 \cos^2 x - b^2 \sin^2 x} = \frac{1}{2ab} \ln \left| \frac{b \tan x + a}{b \tan x - a} \right| + C

  27. ∫xsin⁡axdx=1a2sin⁡ax−1axcos⁡ax+C\int x \sin ax \mathrm{d} x = \frac{1}{a^2} \sin ax - \frac{1}{a} x \cos ax + C

  28. ∫x2sin⁡axdx=−1ax2cos⁡ax+2a2xsin⁡ax+2a3cos⁡ax+C\int x^2 \sin ax \mathrm{d} x = -\frac{1}{a} x^2 \cos ax + \frac{2}{a^2} x \sin ax + \frac{2}{a^3} \cos ax + C

  29. ∫xcos⁡axdx=1a2cos⁡ax+1axsin⁡ax+C\int x \cos ax \mathrm{d} x = \frac{1}{a^2} \cos ax + \frac{1}{a} x \sin ax + C

  30. ∫x2cos⁡axdx=1ax2sin⁡ax+2a2xcos⁡ax−2a3sin⁡ax+C\int x^2 \cos ax \mathrm{d} x = \frac{1}{a} x^2 \sin ax + \frac{2}{a^2} x \cos ax - \frac{2}{a^3} \sin ax + C

(十二) 含有反三角函数的积分 (其中 a>0a > 0)​

  1. ∫arcsin⁡xadx=xarcsin⁡xa+a2−x2+C\int \arcsin \frac{x}{a} \mathrm{d} x = x \arcsin \frac{x}{a} + \sqrt{a^2 - x^2} + C

  2. ∫xarcsin⁡xadx=(x22−a24)arcsin⁡xa+x4a2−x2+C\int x \arcsin \frac{x}{a} \mathrm{d} x = \left( \frac{x^2}{2} - \frac{a^2}{4} \right) \arcsin \frac{x}{a} + \frac{x}{4} \sqrt{a^2 - x^2} + C

  3. ∫x2arcsin⁡xadx=x33arcsin⁡xa+19(x2+2a2)a2−x2+C\int x^2 \arcsin \frac{x}{a} \mathrm{d} x = \frac{x^3}{3} \arcsin \frac{x}{a} + \frac{1}{9} (x^2 + 2a^2) \sqrt{a^2 - x^2} + C

  4. ∫arccos⁡xadx=xarccos⁡xa−a2−x2+C\int \arccos \frac{x}{a} \mathrm{d} x = x \arccos \frac{x}{a} - \sqrt{a^2 - x^2} + C

  5. ∫xarccos⁡xa dx=(x22−a24)arccos⁡xa−x4a2−x2+C\int x \arccos \frac{x}{a} \,\mathrm{d} x = \left( \frac{x^2}{2} - \frac{a^2}{4} \right) \arccos \frac{x}{a} - \frac{x}{4} \sqrt{a^2 - x^2} + C

  6. ∫x2arccos⁡xa dx=x33arccos⁡xa−19(x2+2a2)a2−x2+C\int x^2 \arccos \frac{x}{a} \,\mathrm{d} x = \frac{x^3}{3} \arccos \frac{x}{a} - \frac{1}{9} (x^2 + 2a^2) \sqrt{a^2 - x^2} + C

  7. ∫arctan⁡xa dx=xarctan⁡xa−a2ln⁡(a2+x2)+C\int \arctan \frac{x}{a} \,\mathrm{d} x = x \arctan \frac{x}{a} - \frac{a}{2} \ln \left( a^2 + x^2 \right) + C

  8. ∫xarctan⁡xa dx=12(a2+x2)arctan⁡xa−a2x+C\int x \arctan \frac{x}{a} \,\mathrm{d} x = \frac{1}{2} \left( a^2 + x^2 \right) \arctan \frac{x}{a} - \frac{a}{2} x + C

  9. ∫x2arctan⁡xa dx=x33arctan⁡xa−a6x2+a36ln⁡(a2+x2)+C\int x^2 \arctan \frac{x}{a} \,\mathrm{d} x = \frac{x^3}{3} \arctan \frac{x}{a} - \frac{a}{6} x^2 + \frac{a^3}{6} \ln \left( a^2 + x^2 \right) + C

(十三) 含有指数函数的积分​

  1. ∫ax dx=1ln⁡aax+C\int a^x \,\mathrm{d} x = \frac{1}{\ln a} a^x + C

  2. ∫eax dx=1aeax+C\int e^{ax} \,\mathrm{d} x = \frac{1}{a} e^{ax} + C

  3. ∫xeax dx=1a2(ax−1)eax+C\int x e^{ax} \,\mathrm{d} x = \frac{1}{a^2} (ax - 1) e^{ax} + C

  4. ∫xneax dx=1axneax−na∫xn−1eax dx\int x^n e^{ax} \,\mathrm{d} x = \frac{1}{a} x^n e^{ax} - \frac{n}{a} \int x^{n-1} e^{ax} \,\mathrm{d} x

  5. ∫xax dx=xln⁡aax−1(ln⁡a)2ax+C\int x a^x \,\mathrm{d} x = \frac{x}{\ln a} a^x - \frac{1}{(\ln a)^2} a^x + C

  6. ∫xnax dx=1ln⁡axnax−nln⁡a∫xn−1ax dx\int x^n a^x \,\mathrm{d} x = \frac{1}{\ln a} x^n a^x - \frac{n}{\ln a} \int x^{n-1} a^x \,\mathrm{d} x

  7. ∫eaxsin⁡bx dx=1a2+b2eax(asin⁡bx−bcos⁡bx)+C\int e^{ax} \sin bx \,\mathrm{d} x = \frac{1}{a^2 + b^2} e^{ax} (a \sin bx - b \cos bx) + C

  8. ∫eaxcos⁡bx dx=1a2+b2eax(bsin⁡bx+acos⁡bx)+C\int e^{ax} \cos bx \,\mathrm{d} x = \frac{1}{a^2 + b^2} e^{ax} (b \sin bx + a \cos bx) + C

  9. ∫eaxsin⁡nbx dx=1a2+b2n2eaxsin⁡n−1bx(asin⁡bx−nbcos⁡bx)+n(n−1)b2a2+b2n2∫eaxsin⁡n−2bx dx\int e^{ax} \sin^n bx \,\mathrm{d} x = \frac{1}{a^2 + b^2 n^2} e^{ax} \sin^{n-1} bx \left( a \sin bx - n b \cos bx \right) + \frac{n (n-1) b^2}{a^2 + b^2 n^2} \int e^{ax} \sin^{n-2} bx \,\mathrm{d} x

  10. ∫eaxcos⁡nbx dx=1a2+b2n2eaxcos⁡n−1bx(acos⁡bx+nbsin⁡bx)+n(n−1)b2a2+b2n2∫eaxcos⁡n−2bx dx\int e^{ax} \cos^n bx \,\mathrm{d} x = \frac{1}{a^2 + b^2 n^2} e^{ax} \cos^{n-1} bx \left( a \cos bx + n b \sin bx \right) + \frac{n (n-1) b^2}{a^2 + b^2 n^2} \int e^{ax} \cos^{n-2} bx \,\mathrm{d} x

(十四) 含有对数函数的积分​

  1. ∫ln⁡x dx=xln⁡x−x+C\int \ln x \,\mathrm{d} x = x \ln x - x + C

  2. ∫dxxln⁡x=ln⁡∣ln⁡x∣+C\int \frac{\mathrm{d} x}{x \ln x} = \ln |\ln x| + C

  3. ∫xnln⁡x dx=1n+1xn+1(ln⁡x−1n+1)+C\int x^n \ln x \,\mathrm{d} x = \frac{1}{n+1} x^{n+1} \left( \ln x - \frac{1}{n+1} \right) + C

  4. ∫(ln⁡x)n dx=x(ln⁡x)n−n∫(ln⁡x)n−1 dx\int (\ln x)^n \,\mathrm{d} x = x (\ln x)^n - n \int (\ln x)^{n-1} \,\mathrm{d} x

  5. ∫xm(ln⁡x)n dx=1m+1xm+1(ln⁡x)n−nm+1∫xm(ln⁡x)n−1 dx\int x^m (\ln x)^n \,\mathrm{d} x = \frac{1}{m+1} x^{m+1} (\ln x)^n - \frac{n}{m+1} \int x^m (\ln x)^{n-1} \,\mathrm{d} x

(十五) 含有双曲函数的积分​

  1. ∫sh⁡x dx=ch⁡x+C\int \operatorname{sh} x \,\mathrm{d} x = \operatorname{ch} x + C

  2. ∫ch⁡x dx=sh⁡x+C\int \operatorname{ch} x \,\mathrm{d} x = \operatorname{sh} x + C

  3. ∫th⁡x dx=ln⁡ch⁡x+C\int \operatorname{th} x \,\mathrm{d} x = \ln \operatorname{ch} x + C

  4. ∫sh⁡2x dx=−x2+14sh⁡2x+C\int \operatorname{sh}^2 x \,\mathrm{d} x = -\frac{x}{2} + \frac{1}{4} \operatorname{sh} 2x + C

  5. ∫ch⁡2x dx=x2+14sh⁡2x+C\int \operatorname{ch}^2 x \,\mathrm{d} x = \frac{x}{2} + \frac{1}{4} \operatorname{sh} 2x + C

(十六) 定积分​

  1. ∫−ππcos⁡nx dx=∫−ππsin⁡nx dx=0\int_{-\pi}^{\pi} \cos nx \,\mathrm{d} x = \int_{-\pi}^{\pi} \sin nx \,\mathrm{d} x = 0

  2. ∫−ππcos⁡mxsin⁡nx dx=0\int_{-\pi}^{\pi} \cos mx \sin nx \,\mathrm{d} x = 0

  3. ∫−ππcos⁡mxcos⁡nx dx={0,m≠n,π,m=n.\int_{-\pi}^{\pi} \cos mx \cos nx \,\mathrm{d} x = \begin{cases} 0, & m \neq n, \\ \pi, & m = n. \end{cases}

  4. ∫−ππsin⁡mxsin⁡nx dx={0,m≠n,π,m=n.\int_{-\pi}^{\pi} \sin mx \sin nx \,\mathrm{d} x = \begin{cases} 0, & m \neq n, \\ \pi, & m = n. \end{cases}

  5. ∫0πsin⁡mxsin⁡nx dx=∫0πcos⁡mxcos⁡nx dx={0,m≠n,π2,m=n.\int_{0}^{\pi} \sin mx \sin nx \,\mathrm{d} x = \int_{0}^{\pi} \cos mx \cos nx \,\mathrm{d} x = \begin{cases} 0, & m \neq n, \\ \frac{\pi}{2}, & m = n. \end{cases}

  6. In=∫0π2sin⁡nx dx=∫0π2cos⁡nx dxI_n = \int_{0}^{\frac{\pi}{2}} \sin^n x \, \mathrm{d} x = \int_{0}^{\frac{\pi}{2}} \cos^n x \, \mathrm{d} x

    In=n−1nIn−2I_n = \frac{n-1}{n} I_{n-2}

    ={n−1n⋅n−3n−2⋅ ⋯ ⋅45⋅23(n 为大于 1 的正奇数),I1=1,n−1n⋅n−3n−2⋅ ⋯ ⋅34⋅12⋅π2(n 为正偶数),I0=π2.= \begin{cases} \frac{n-1}{n} \cdot \frac{n-3}{n-2} \cdot \, \cdots \, \cdot \frac{4}{5} \cdot \frac{2}{3} & (n \text{ 为大于 }1\text{ 的正奇数}), & I_1 = 1, \\ \frac{n-1}{n} \cdot \frac{n-3}{n-2} \cdot \, \cdots \, \cdot \frac{3}{4} \cdot \frac{1}{2} \cdot \frac{\pi}{2} & (n \text{ 为正偶数}), & I_0 = \frac{\pi}{2}. \end{cases}