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The only method to find the original function if you know the integrand is by the use of integration. This is because differentiation and integration are inverse functions. Though the process of integration does not allow us to identify constants which are eliminated during differentiation, it is the only choice.
The only method to find the primitive when knowing the integrand is to determine the integral of the integrand.
For instance, we'll have f'(x) = 49x^6-3x^2 and we'll determine F(x).
We'll apply the additive property of integrals:
Int (49x^6-3x^2)dx = Int (49x^6)dx - Int (3x^2)dx
We'll re-write the sum of integrals, taking out the constants:
Int (49x^6-3x^2)dx = 49Int x^6 dx - 3Int x^2 dx
Int (49x^6-3x^2)dx = 49*x^7/7 - 3*x^3/3
We'll simplify and we'll get:
Int (49x^6-3x^2)dx = 7x^7 - x^3 + C
The primitive function F(x) is: F(x) = 7x^7 - x^3 + C
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