Collection of Maths Problems

Integrals of rational functions

Task number: 3111

• Variant 1

  \( \int \frac{3x + 5}{2x^2 + 3x + 7} \, dx \)

• Result

  \[ \frac{11}{2\sqrt{47}} \arctan \left( \frac{4x + 3}{\sqrt{47}} \right) + \frac34 \ln(2x^2 + 3x + 7) + C \]

• Variant 2

  \( \int\frac{x^7 - 5}{x^2 - 1} \, dx \)

• Result

  \[ \frac{x^6}6 + \frac{x^4}4 + \frac{x^2}2 + 3 \ln|x+ 1| - 2 \ln|x-1| + C, \] where the constant \(C\) can have a different value on each open interval \((-\infty, -1)\), \((-1, 1)\) and \((1, \infty)\)

• Variant 3

  \( \int \frac{x^3 + 1}{x^3 - 5x^2 + 6x} \, dx \)

• Result

  \[ x + \frac{\ln|x|}6- \frac92 \ln|x-2| + \frac{28}3 \ln|x-3| + C, \] where the constant \(C\) can have a different value on each open interval \((-\infty, 0)\), \((0, 2)\), \((2, 3)\) and \((3, \infty)\)

• Variant 4

  \( \int \frac{x^4}{x^4 + 5x^2 + 4} \, dx \)

• Result

  \[ x + \frac{\arctan x}3 - \frac83 \arctan \left(\frac x2 \right) + C. \] Remark: The solution can be in a different form and it may not be readily apparent that the given expressions are equivalent. For example \[ x + \frac{\arctan x}3 + \frac83 \arctan \left(\frac 2x \right) + C \] is also correct, because \[ \arctan \left(\frac 2x \right) = - \arctan \left(\frac x2 \right) + \frac{\pi}2 \]

• Variant 5

  \( \int\frac{x}{x^3 - 3x + 2} \, dx \)

• Result

  \[ - \frac{1}{3(x-1)} + \frac29 \ln|x- 1| - \frac29 \ln|x+2| + C, \] where the constant \(C\) can have a different value on each open interval \((-\infty, -2)\), \((-2, 1)\) and \((1, \infty)\)

• Variant 6

  \( \int \frac{x}{x^3 - 1} \, dx \)

• Variant 7

  \( \int \frac{1}{x^6 + 1} \, dx \)

• Variant 8

  \( \int \frac{x}{2x^2 - 3x + 2} \, dx \)

• Variant 9

  \( \int \frac{x^3 + 1}{x^3 - x^2} \, dx \)

• Variant 10

  \( \int \frac{x^2}{(x+2)^2(x+4)^2} \, dx \)