Short and Sweet Calculus

## 5.1Definition

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 $$f$$↩︎