Math topic中文

Partial fractions

partial fractions with cases where the denominator is no more complicated than:

  • (ax+b)(cx+d)(ax + b)(cx + d)
  • (ax+b)(cx+d)2(ax + b)(cx + d)^{2}
  • (ax+b)(x2+c2)(ax + b)(x^{2} + c^{2})

Practice questions

  1. 1Express 2x4+x3+9x2+7x+3(x+1)(x2+5)\frac{2x^4 + x^3 + 9x^2 + 7x + 3}{(x+1)(x^2+5)} in partial fractions.
  2. 2Express x2+3x+2x2(2x+1)\dfrac{x^2+3x+2}{x^2(2x+1)} in partial fractions. Hence integrate x2+3x+2x2(2x+1)\dfrac{x^2+3x+2}{x^2(2x+1)} with respect t…
  3. 3Express 8x+13(1+2x)(2+x)2\frac{8x+13}{(1+2x)(2+x)^2} in partial fractions. Express 8x+13(1+2x)(2+x)2\frac{8x+13}{(1+2x)(2+x)^2} in partial fractions…
  4. 4Express 2x2+3xx2+3x+2\dfrac{2x^{2}+3x}{x^{2}+3x+2} in partial fractions. Hence evaluate ∫132x2+3xx2+3x+2 dx\displaystyle \int_{1}^{3} \dfrac{2x^{2}+3x}{x^{2}+3x+2}\,dx…
  5. 5Express 3x2+5x−2(x+1)(x2+4)\frac{3x^2 + 5x - 2}{(x + 1)(x^2 + 4)} in partial fractions.
  6. 6Express 2x+15(3+x)(3−2x)\frac{2x+15}{(3+x)(3-2x)} in partial fractions. Hence evaluate ∫012x+15(3+x)(3−2x) dx\int_{0}^{1} \frac{2x+15}{(3+x)(3-2x)} \, dx.
  7. 7Express 5x3−4x2+15x−21(x−1)(x2+4)\frac{5x^3 - 4x^2 + 15x - 21}{(x-1)(x^2 + 4)} in partial fractions.
  8. 8Express x+7(x2−9)(x−3)\frac{x + 7}{(x^2 - 9)(x - 3)} as the sum of 3 partial fractions.
  9. 9Express −3x2+12x−1(3x+1)(x−1)2\frac{-3x^2 + 12x - 1}{(3x + 1)(x - 1)^2} in partial fractions.
  10. 10Find the value of AA, BB, CC, and DD for which 4x3+7x2−13x−2(x2+3)(x−1)=A+Bx+Cx2+3+Dx−1\frac{4x^3 + 7x^2 - 13x - 2}{(x^2 + 3)(x - 1)} = A + \frac{Bx + C}{x^2 + 3} + \frac{D}{x - 1}…
  11. 11Express 8−3x(2x+3)(x2+4)\dfrac{8 - 3x}{(2x + 3)(x^2 + 4)} in partial fractions. Differentiate ln⁡(x2+4)\ln\left(x^2 + 4\right) with respect t…
  12. 12Express 2x3−5x2(x−1)\frac{2x^3 - 5}{x^2(x - 1)} in partial fractions.
  13. 13Show that x−2x - 2 is a factor of x3−12x+16x^3 - 12x + 16. Express 2+x2x3−12x+16\frac{2 + x^2}{x^3 - 12x + 16} as the sum of three partia…
  14. 14Express 13x−6x2(2x−3)\frac{13x - 6}{x^2(2x - 3)} in partial fractions. Hence ∫13x−6x2(2x−3) dx\int \frac{13x - 6}{x^2(2x - 3)} \, dx.
  15. 15Show that x−2x - 2 is a factor of x3−2x2+3x−6x^{3} - 2x^{2} + 3x - 6. Express 11−2xx3−2x2+3x−6\dfrac{11 - 2x}{x^{3} - 2x^{2} + 3x - 6} as partia…
  16. 16By using long division, show that 2x2+12x^2+1 is a factor of 2x3−4x2+x−22x^3-4x^2+x-2. Express −5x2+11x−112x3−4x2+x−2\dfrac{-5x^2+11x-11}{2x^3-4x^2+x-2}…
  17. 17The equation of a curve is y=1−sin⁡xcos⁡xy = \frac{1 - \sin x}{\cos x}. Show that dydx=tan⁡xsec⁡x−sec⁡2x\frac{dy}{dx} = \tan x \sec x - \sec^2 x. Hence…
  18. 18Express 3x2−4x2(3x−2)\frac{3x^2 - 4}{x^2(3x - 2)} in partial fractions. Hence, evaluate ∫3x2−4x2(3x−2) dx\int \frac{3x^2 - 4}{x^2(3x - 2)} \, dx.
  19. 19Factorise x4−1x^4 - 1 completely. Given that x4x4−1=a+bx4−1\frac{x^4}{x^4 - 1} = a + \frac{b}{x^4 - 1}, find the values of aa and of…
  20. 20Express 9x2+17x−163x2+5x−2\frac{9x^2 + 17x - 16}{3x^2 + 5x - 2} in partial fractions. Hence, find ∫9x2+17x−163x2+5x−2 dx\int \frac{9x^2 + 17x - 16}{3x^2 + 5x - 2} \, dx…
  21. 21Express 8x3+3x+1x(2x+1)2\frac{8x^3 + 3x + 1}{x(2x + 1)^2} as a sum of a polynomial and a proper fraction. Hence, express 8x3+3x+1x(2x+1)2\frac{8x^3 + 3x + 1}{x(2x + 1)^2}…
  22. 22Express 2x3+x2+x(x−1)(x2+1)\frac{2x^3 + x^2 + x}{(x-1)(x^2 + 1)} in partial fractions. Differentiate ln⁡(x2+1)\ln(x^2 + 1) with respect to xx.…
  23. 23Given that f(x)=x3+3x2−4\text{f}(x) = x^3 + 3x^2 - 4, show that x+2x+2 is a factor of f(x)\text{f}(x), and hence factorise f(x)\text{f}(x)…
  24. 24Factorise completely 2x3+7x2+4x−42x^3+7x^2+4x-4. Express 12x2+32x+312x3+7x2+4x−4\dfrac{12x^2+32x+31}{2x^3+7x^2+4x-4} in partial fractions.
  25. 25Express 2x3−4x2−5x+3(x−3)(x2−9)\frac{2x^3 - 4x^2 - 5x + 3}{(x-3)(x^2-9)} in partial fractions.
  26. 26Factorise 3x3−24y33x^3 - 24y^3 completely. Express 7x2+19x+15(x+1)2(x+2)\frac{7x^2 + 19x + 15}{(x + 1)^2(x + 2)} as partial fractions.
  27. 27Express 10x3−5x\frac{10x}{3-5x} in the form a+b3−5xa + \frac{b}{3-5x}, where aa and bb are integers. Hence, integrate 10x3−5x\frac{10x}{3-5x}…
  28. 28By using long division, divide 4x3+5x2+x−14x^3 + 5x^2 + x - 1 by x2(x+1)x^2 (x+1). Hence, express 4x3+5x2+x−1x2(x+1)\dfrac{4x^3 + 5x^2 + x - 1}{x^2 (x+1)}…
  29. 29Express x2+1x2−1\frac{x^2 + 1}{x^2 - 1} in partial fractions.
  30. 30Express 13x−6x2(2x−3)\frac{13x - 6}{x^2(2x - 3)} in partial fractions.
  31. 31Express 14x2−15x−29(x−1)2(2x+3)\frac{14x^2 - 15x - 29}{(x-1)^2(2x+3)} in partial fractions.
  32. 32Express 7x+8(2x+1)(x−1)2\frac{7x + 8}{(2x + 1)(x - 1)^2} in partial fractions.
  33. 33Express 8x2−x4x2−1\frac{8x^2 - x}{4x^2 - 1} in partial fractions. Use your results in part (i) to find ∫8x2−x4x2−1 dx\int \frac{8x^2 - x}{4x^2 - 1} \, dx…
  34. 34Express 4x3−x2+37x−50(x2+9)(x−1)\frac{4x^3-x^2+37x-50}{(x^2+9)(x-1)} in partial fractions.
  35. 35Express −x2+2x+1(x−1)(x2+1)\frac{-x^2 + 2x + 1}{(x-1)(x^2 + 1)} in the form Ax−1+Bx+Cx2+1\frac{A}{x-1} + \frac{Bx + C}{x^2 + 1} where AA, BB and CC…
  36. 36Express 2x3−4x2−5x+3(x−3)(x2−9)\frac{2x^3 - 4x^2 - 5x + 3}{(x - 3)(x^2 - 9)} in partial fractions. Hence find ∫2x3−4x2−5x+3(x−3)(x2−9) dx\int \frac{2x^3 - 4x^2 - 5x + 3}{(x - 3)(x^2 - 9)} \, dx…
  37. 37Express 2x−1x2(x+1)\frac{2x-1}{x^2(x+1)} in partial fractions.
  38. 38Express 2x3+2x2−7x+4x(x−1)2\frac{2x^3+2x^2-7x+4}{x(x-1)^2} in partial fractions. Hence evaluate ∫244x3+4x2−14x+83x(x−1)2 dx\int_{2}^{4} \frac{4x^3+4x^2-14x+8}{3x(x-1)^2} \, dx…
  39. 39Express 15x2−13x+18(x−2)(3x2+1)\frac{15x^2 - 13x + 18}{(x - 2)(3x^2 + 1)} in partial fractions.