Integral Calculation and Partial Fraction Decomposition

Solving the Integral ( \int \frac{16x}{16x^4} \, dx ) Using Partial Fraction Decomposition

Step 1: Simplifying the Integral

• Start with the integral:
  • [ S = \int \frac{16x}{16x^4} \, dx ]
• Simplify the integrand:
  • [ S = \int \frac{16x}{16x^4} \, dx = \int \frac{1}{x^3} \, dx ]
  • This simplification directly follows from dividing the numerator and denominator by ( 16 ).

Step 2: Partial Fraction Decomposition

• The integral simplifies to:
  • [ S = \int x^{-3} \, dx ]

Step 3: Finding the Antiderivative

• To find the antiderivative of ( x^{-3} ):
  • Use the power rule, which states:
  • If ( n \neq -1 ), ( \int x^n \, dx = \frac{x^{n+1}}{n+1} + C )
• Applying the power rule:
  • Let ( n = -3 ), hence:
  • [ \int x^{-3} \, dx = \frac{x^{-3+1}}{-3+1} + C ]
  • [ = \frac{x^{-2}}{-2} + C ]
  • [ = -\frac{1}{2x^2} + C ]

Step 4: Including Absolute Values

• Remember to consider absolute values in logarithmic integrals:
  • If applicable, when applying logarithms, include ( +C ) and ensure absolute values are used:
  • If the result had included a logarithmic term, it would look like:
  • [ \int \frac{1}{x} \, dx = \ln |x| + C ]

Final Result

• Therefore, the solution to the integral ( \int \frac{16x}{16x^4} \, dx ) is:
  • [ S = -\frac{1}{2x^2} + C ]
• In more complex integrals involving logarithmic functions, the common format should use absolute values as needed.