Let us study the properties of indefinite integrals to work on them.
- The derivative of an integral is the integrand itself. ∫ f(x) dx = f(x) +C
- Two indefinite integrals with the same derivative lead to the same family of curves and so they are equivalent. ∫ [ f(x) dx -g(x) dx] =0
- The integral of the sum or difference of a finite number of functions is equal to the sum or difference of the integrals of the individual functions. ∫ [ f(x) dx+g(x) dx] = ∫ f(x) dx + ∫ g(x) dx
- The constant is taken outside the integral sign. ∫ k f(x) dx = k ∫ f(x) dx, where k ∈ R.
- The previous two properties are combined to get the form: ∫ [k11f11(x) + k22f22(x) +… knnfnn(x)] dx = k11∫ f11(x)dx + k22∫ f22(x)dx+ … knn ∫ fnn(x)dx