Indefinite Integrals as Antiderivative

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Consider the following example:

Let f(x) = cos 3x, let us find a function F(x) such that

We know that

Here

In other words we say the integral cos 3x is

Suppose then also we have

Let us define integral of a function in general as follows.

Let F(x) be a function such that

\ In general, integral of f(x) is F(x) + C

Where C is called the constant of integration.

List of Standard Integrals

Geometrical Interpretation of indefinite integral

Let f(x) = 3x²

Note that for different values of C we get different integrals. But all these integrals are very similar geometrically.

The function y = x³ + C represent a family of integrals. The above figure shows different curves of the integral function y = x³ + C. These curves fill the co-ordinate plane without overlapping. These curves together constitute the indefinite integrals.

If we draw a line x = a perpendicular to X-axis. Then the curves y = x³ + C have slopes The slopes of the tangent at P₁, P₂, P₃, P₄ and P₅ are equal (equal to 3a²). This indicates, the tangents to these curves are parallel at these points.