Functions defined using integrals

Not all integrals have antiderivatives that can be written down using the basic functions and operations that we know. Some functions are, in fact, defined using an integral. This page shows two such functions.

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1. Sine integral

The applet shows a graph on the left of (sin x)/x and on the right the graph of This function, defined using an integral, is called the Sine Integral and frequently written as Si(x). There is no antiderivative for the integrand that can be written using the basic operations of algebra and the basic functions we know; the only way to define the sine integral is using this integral. Move the x slider to see various values of the function and the area that is accumulated to generate that value.

2. Error function

Select the second example, showing the error function, This is another function that is defined using an integral. Move the x slider to see various values of the function and the area that is accumulated to generate that value.

Other 'Constructing Antiderivatives' topics

• Antiderivatives from Slope and Indefinite Integral
• Accumulation Functions
• Basic Antiderivatives
• Introduction to Differential Equations
• Second Fundamental Theorem of Calculus
• Functions Defined Using Integrals
• Equations of Motion