Integration of Rational Functions

In Topic 6, Differentiation of Logarithmic Functions it was shown that if f(x) = ln x then f '(x) = Y12_Integration_of_Rational_Functions_01.gif.

Because integration is the same process as anti-differentiation it follows that:

 Y12_Integration_of_Rational_Functions_02.gif = ln IxI + c.

The absolute (positive) value of x is used because the function y = ln x is only defined for positive values of x.

More complex functions are integrated by inspection.


∫ Y12_Integration_of_Rational_Functions_03.gif dx = 3 ∫ Y12_Integration_of_Rational_Functions_01.gif dx = 3 ln IxI + c.

∫ Y12_Integration_of_Rational_Functions_04.gif dx = Y12_Integration_of_Rational_Functions_05.gif ln I2x + 3I + c

Integration of Rational Functions

When finding integrals such as Y12_Integration_of_Rational_Functions_06.gif and other expressions in fractional form it is sometimes best to divide first.