(a) In Section 2.8 we defined an antiderivative of f to be a...

(a) In Section 2.8 we defined an antiderivative of $f$ to be a function $F$ such that $F^{\prime}=f .$ Try to guess a formula for an antiderivative of $f(x)=x^{2} .$ Then check your answer by differentiating it. How many antiderivatives does $f$ have? (b) Find antiderivatives for $f(x)=x^{3}$ and $f(x)=x^{4}.$ (c) Find an antiderivative for $f(x)=x^{n}$, where $n \neq-1$ Check by differentiation.

(a) In Section 2.8 we defined an antiderivative of $f$ to be a function $F$ such that $F^{\prime}=f .$ Try to guess a formula for an antiderivative of $f(x)=x^{2} .$ Then check your answer by differentiating it. How many antiderivatives does $f$ have?
(b) Find antiderivatives for $f(x)=x^{3}$ and $f(x)=x^{4}.$
(c) Find an antiderivative for $f(x)=x^{n}$, where $n \neq-1$ Check by differentiation.

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Key Concepts

Antiderivative

An antiderivative of a function f is another function F whose derivative is f. This concept forms the basis of finding indefinite integrals in calculus, where differentiating F returns the original function. It is essential to understand that any two antiderivatives of the same function differ by a constant.

Constant of Integration

When finding the antiderivative of a function, the result includes an arbitrary constant (commonly written as C) because the derivative of a constant is zero. This constant represents the family of solutions, meaning there are infinitely many antiderivatives for a given function.

Power Rule for Integration

The power rule for integration is a fundamental technique for finding antiderivatives of functions of the form x^n, where n is any real number except -1. The rule states that the antiderivative is x^(n+1)/(n+1) plus a constant of integration. This rule is central to solving problems involving polynomial functions.

Verification by Differentiation

After finding a candidate antiderivative, verifying it by differentiation is important. This process involves differentiating the proposed antiderivative to ensure that it yields the original function. This check confirms the accuracy of the integration process.

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