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Find {eq}f {/eq}, if {eq}f'(x) = x^4 - 5x^2 + 3x - 7 {/eq}, {eq}f(0) = 2 {/eq}.

Question:

Find {eq}f {/eq}, if {eq}f'(x) = x^4 - 5x^2 + 3x - 7 {/eq}, {eq}f(0) = 2 {/eq}.

Antiderivative:

As the name suggests, antiderivative is the opposite of the derivative. To determine the antiderivative of a function, integration is performed. Therefore, integral can be considered as the antiderivative. For indefinite integral, the final answer contains a constant of integration. If a condition is given, then the value of the integral constant can be determined. The reverse power rule is used to determine the integral of simple polynomial expressions. It is the reverse process of the power rule of the derivatives. It can be used as:

{eq}\int x^a \ dx = \dfrac{x^{a+1}}{a+1} + C {/eq}

Answer and Explanation: 1

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Given:

{eq}f'(x) = x^4 - 5x^2 + 3x - 7 {/eq}

Integrating the derivative:

{eq}\int f'(x) dx = \int (x^4 - 5x^2 + 3x - 7) \ dx \\ \Rightarrow...

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Finding Antiderivatives Using Initial Conditions
Finding Antiderivatives Using Initial Conditions

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Chapter 26 / Lesson 2
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This lesson will briefly review antiderivatives, and then we will introduce a step by step algorithm that can be used to find antiderivatives using initial conditions. We will apply the algorithm in various examples.


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