Question

Find an antiderivative. $$p(t)=t^{3}-\frac{t^{2}}{2}-t$$

   Find an antiderivative.
$$p(t)=t^{3}-\frac{t^{2}}{2}-t$$

Applied Calculus

Applied Calculus

Deborah… 5th Edition

Chapter 6, Problem 29

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Solved by Expert Linh Vu

07/10/2020

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The power rule states that the antiderivative of $t^n$ is $\frac{t^{n+1}}{n+1}$. Show more…

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Find an antiderivative. $$p(t)=t^{3}-\frac{t^{2}}{2}-t$$

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Key Concepts

Constant of Integration

When finding an antiderivative, the constant of integration is included because differentiation of a constant gives zero. This constant represents the fact that there is a whole family of functions that differ only by a constant, all of which serve as valid antiderivatives.

Linearity of Integration

The linearity of integration is a fundamental property which states that the integral of a sum of functions is the sum of their integrals, and a constant multiplier can be taken outside the integral. This property simplifies the process of integrating polynomials by allowing each term to be integrated separately.

Antiderivative (Indefinite Integral)

An antiderivative of a function is another function whose derivative equals the original function. In calculus, finding an antiderivative is equivalent to computing the indefinite integral of the function, and it represents a family of functions that differ by a constant.

Power Rule for Integration

The power rule for integration is a basic technique used when integrating functions of the form t^n, where n is a real number not equal to -1. It states that the integral of t^n is t^(n+1)/(n+1) plus a constant of integration, which is used to handle each term when integrating polynomials.

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