Find a function f(x), such that {eq}f'(x) = x ^{3} + 2 {/eq} and f(0) = 0.

Question:

Find a function f(x), such that {eq}f'(x) = x ^{3} + 2 {/eq} and f(0) = 0.

Rule for Power and Constant:

For the anti-derivative of algebraic expression {eq}x^m+A {/eq}, we have to use the power rule of integration for the term {eq}x^m {/eq} and rule for a constant value A so that we can compute the anti-derivative of each term individually. The required mathematical expression is:

{eq}\begin{align*} \displaystyle \int x^m\ dx&=\frac{x^{m+1}}{m+1}+C\\ \displaystyle \int A\ dx&=Ax+C\\ \displaystyle \int(x^m+ A)\ dx&=\frac{x^{m+1}}{m+1}+Ax+C \end{align*} {/eq}, where,

  • A and m are constant values.

Answer and Explanation: 1

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Given derivative:

The 1st order derivative function with the initial value is:

{eq}f'(x) = x ^{3} + 2\\ f(0) = 0 {/eq}

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Anti-Derivatives: Calculating Indefinite Integrals of Polynomials

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Chapter 13 / Lesson 2
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The fundamental theorem of calculus allows us to calculate indefinite integrals as the anti-derivatives of the original polynomial function. Learn how to calculate indefinite integrals of polynomials through several examples and how to apply a general rule to polynomials with any number of variables.


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