Short and Sweet Calculus

5.1 Definition

Differentiation is the process of finding the derivative of a given function. Sometimes we need to reverse the process of differentiation. That is, we want to discover an unknown function whose derivative is known.

  • If \(f(x)\) is given, a function \(F\) such that \[F'(x)=f(x)\] is called an antiderivative or an integral8 of \(f\) and the process of finding \(F(x)\) from \(f(x)\) is called antidifferentiation or integration.

For example, because \(\dfrac{d}{dx}x^{3}=3x^{2}\), \(F(x)=x^{3}\) is an integral of \(f(x)=3x^{2}\). Because the derivative of a constant is zero, the derivatives of the following functions \[x^{3}+1,\quad x^{3}-\pi,\quad x^{3}+\sqrt{5}\] and in general \[x^{3}+C\] are the same.

  • In general, if \(F(x)\) is an integral of \(f(x)\), then \(F(x)+C\) is also an integral of \(f(x)\), where \(C\) is any arbitrary number.

  • If \(F(x)\) is a function whose derivative is \(f(x)\) over an interval, then all functions having the same derivative \(f(x)\) are of the form \(F(x)+C\), where \(C\) is a constant. In other words, \(f\) does not have an integral (or antiderivative) that cannot be written as \(F(x)+C\) for some constant \(C\).

5.1. Let \(F(x)\) and \(G(x)\) be two functions such that \[F'(x)=G'(x),\] on a certain interval, then there exist a constant \(C\) such that \[F(x)=G(x)+C,\] for all \(x\) in the interval.

Proof. Let \(h(x)=F(x)-G(x)\). Then \(h'(x)=F'(x)-G'(x)=0\). This shows that the slope of the tangent line to the graph of \(h\) at each point is zero and tangents are always horizontal. Therefore, the graph of \(h\) cannot go up or down . Therefore, \(h\) is a constant function \(h(x)=C\) or \(F(x)=G(x)+C\). ◻

If \(\dfrac{dF(x)}{dx}=f(x)\) or equivalently if \(dF(x)=f(x)dx\), we write \[\int f(x)dx=F(x)+C,\] where \(C\) is called the constant of integration. Because \(C\) is unknown and indefinite, \(\int f(x)dx\) is called the indefinite integral of \(f(x)\).

  • The symbol \(\int\) is called the integral sign.

  • \(f(x)\) is called the integrand or subject of integration.

  • The differential \(dx\) indicates that \(x\) is the variable of integration.

For example, because \(\dfrac{d}{dx}\sin x=\cos x\) or \(d(\sin x)=\cos x\:dx\), we have \[\int\cos x\:dx=\sin x+C.\] Also because \(d(x^{3})=3x^{2}dx\), we have \[\int3x^{2}dx=x^{3}+C.\]

  • Notice that \[\boxed{\frac{d}{dx}\left[\int f(x)dx\right]=f(x).}\]

  • We say a function is integrable if its integral exists. All continuous function are integrable.

  1. We may also say that \(F\) is an integral function of \(f\)↩︎

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