Antiderivatives and Indefinite Integrals

Definition

Definition (Antiderivative)
 
A function \( F \) is an antiderivative of a function \( f \) on the interval \( (a,b) \) if \( F^{\prime}(x)=f(x) \) for every \( x\in(a,b) \).

The function \( f(x)=3x^2 \) has many anti-derivatives: \( F_1(x)=x^3 \), \( F_2(x)=x^3-17 \), \( F_3(x)=x^3+41 \), etc. They all differ by a constant factor.

Definition (Indefinite integral)
 
The set of all antiderivatives of a given function \( f \) is called the indefinite integral of \( f \), and it is denoted as \( \int f(x)\,dx \). More precisely \[ \int f(x)\,dx=\left\{ F: F^{\prime}(x)=f(x)\right\}.\]

Consider the function \( f(x)=\cos x \). For each real number \( C \), the function \( F_C(x)=\sin x+C \) is an antiderivative of \( f \). We write \[ \int \cos x\,dx=\sin x+C.\]

Main properties of indefinite integrals

Theorem
 
If \( F_1 \) and \( F_2 \) are two antiderivatives of \( f \) then there exists a real number \( C \) such that \( F_1(x)-F_2(x)=C \) for all \( x \).

The following theorem is easy to prove using the main properties of derivatives.

Theorem
 

For any function \( f \) and any real number \( \alpha \) the following identity holds: \[ \int \alpha \cdot f(x)\,dx=\alpha\int f(x)\,dx.\]

For any two functions \( f \) and \( g \) the following identity holds: \[ \int (f+g)(x)\,dx=\int f(x)\,dx+\int g(x)\,dx.\]

We can use the previous theorem to find anti-derivatives of polynomials. For example, if \( P(x)=x^3-2x^2+11x+4 \), then \[ \int P(x)\,dx=\int x^3\,dx-2\int x^2\,dx+11\int x\,dx+4\int 1\,dx= \frac14x^3-2\frac{x^3}3+11\frac{x^2}2+4x+C,\] where \( C \) could be any real number.


2005-2017 IMOmath.com | imomath"at"gmail.com | Math rendered by MathJax
Home | Olympiads | Book | Training | IMO Results | Forum | Links | About | Contact us