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.

Answer and Explanation: 1

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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 (...

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Antiderivative: Rules, Formula & Examples

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Chapter 8 / Lesson 12
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Understand what an antiderivative is and what antiderivative rules are. Use various antiderivative formulas and learn how to do antiderivatives. See the antiderivative chart for common functions and practice solving basic antiderivatives examples.


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