Properties of the Indefinite Integral

1.  ∫ k f ( x ) dx = k ∫ f ( x ) dx where k refer for any number.  Thus, we can factor multiplicative constants out of indefinite integrals.

2.  ∫ - f ( x ) dx = -∫ f ( x ) dx .  It is really the first property with k = -1 and therefore no proof of this property will be given.

3.  ∫ f (x) ± g (x) dx = ∫ f (x) dx ± ∫ g (x) dx .  In other terms, the integral of a sum or difference of any functions is the sum or difference of the individual integrals. This rule can be extended to as several functions as we required.

There is one last topic to be discussed in this section.  On instance we will be given f ′ ( x ) and will ask f ( x ) was. Now we can answer this question simply with an indefinite integral

                                                          f ( x ) = ∫ f ′ ( x ) dx

Example  If f ′ ( x ) = x⁴ + 3x - 9 what was f ( x ) ?

Solution: By this point it is a simple question to answer.

f ( x ) = ∫f ′ ( x ) dx =  x⁴ + 3x - 9 dx = 1/5 x⁵ + 3/2 x² - 9x + c