In Topic 6, Differentiation of Logarithmic Functions it was shown that if f(x) = ln x then f '(x) = .
Because integration is the same process as anti-differentiation it follows that:
∫ = 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.
∫ dx = 3 ∫ dx = 3 ln IxI + c.
∫ dx = ln I2x + 3I + c
Integration of Rational Functions
When finding integrals such as and other expressions in fractional form it is sometimes best to divide first.