Rules of Integration
The following rules of integration are obtained by reversing the corresponding
rules of differentiation. Their accuracy is easily checked, since the derivative of
the integral must equal the integrand. Each rule is illustrated in Example 2 and
Problems 14.1 to 14.6
Rule 1. The integral of a constant k is
โซ ๐ ๐๐ฅ = ๐๐ฅ + ๐
Rule 2. The integral of 1, written simply as ๐๐ฅ, not 1๐๐ฅ a constant k is
โซ ๐ ๐๐ฅ = ๐ฅ + ๐
Rule 3. The integral of a power function ๐ฅ๐, where ๐ โ โ1, is given by the power
rule
โซ๐ฅ๐๐๐ฅ = 1
๐+ 1 ๐ฅ๐+1 + ๐ถ ๐ โ โ1
Rule 4. The integral of ๐ฅโ1 (or 1/๐ฅ) is
โซ๐ฅโ1๐๐ฅ =ln๐ฅ + ๐ถ ๐ฅ > 0
The condition ๐ฅ > 0 is added because only positive numbers have logarithms. For negative
numbers,
โซ๐ฅโ1๐๐ฅ =ln|๐ฅ|+ ๐ถ ๐ฅ โ 0
Rule 5. The integral of an exponential function is
โซ๐๐๐ฅ๐๐ฅ = ๐๐๐ฅ
๐ln๐ + ๐ถ
Rule 6. The integral of a natural exponential function is
โซ๐๐๐ฅ๐๐ฅ = ๐๐๐ฅ
๐ + ๐ถ since ln๐ = 1
Rule 7. The integral of a constant times a function equals the constant times the integral
of the function.
โซ๐๐(๐ฅ)๐๐ฅ = ๐ โซ ๐(๐ฅ)๐๐ฅ
Rule 8. The integral of the sum or difference of two or more functions equals the sum or
difference of their integrals
โซ[๐(๐ฅ)ยฑ๐(๐ฅ)]๐๐ฅ = โซ ๐(๐ฅ)๐๐ฅ ยฑ โซ๐(๐ฅ) ๐๐ฅ
