Find {eq}\displaystyle f {/eq} for the following function. {eq}\displaystyle f''(x) = 20x^3-12x^2+6x {/eq}

Question:

Find {eq}\displaystyle f {/eq} for the following function.

{eq}\displaystyle f''(x) = 20x^3-12x^2+6x {/eq}

Anti-derivative:

When a function is differentiated, the derivative is obtained. If the function is differentiated just once, the derivative is known as the first derivative of the function. On differentiating the first-order derivative once more, we get the double derivative. If we are given the double-derivative, to obtain the function, we have to apply the process opposite to differentiation. This process is known as the anti-derivative of the function. The anti-derivative is basically indefinite integral of the function. Some of the useful formulae for the calculation of anti-derivative are given below:

{eq}* \int (f(x) \pm g(x)) \ dx = \int f(x) \ dx \pm \int g(x) \ dx \\ * \int K f(x) \ dx = K \int f(x) \ dx \\ * \int x^n \ dx =\dfrac {x^{n+1}}{n+1} +C \\ {/eq}

where '{eq}C {/eq}' is the constant of integration.

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Given:

{eq}\displaystyle f''(x) = 20x^3-12x^2+6x \\ {/eq}

Integrating the double-derivative:

{eq}\displaystyle \int f''(x) \ dx = \int (...