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}
Here, the...
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Chapter 13 / Lesson 2The 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.