It is the branch of calculus where we study about integrals and their properties. Integration is a very important concept. It is mostly useful for following two purposes:

1. To calculate f from f’. If a function f is differentiable in the interval of consideration, then f’ is defined in that interval. We have already seen in differential calculus how to calculate derivatives of a function. We can “undo” that with the help of integral calculus.

2. To calculate area under a curve.

Until now, we have learned that areas are always positive. But as a matter of fact, there is something called signed area.

Let us now go ahead and understand integration meaning.

Integration:

If we know the f’ of a function which is differentiable in its domain, we can then calculate f. In differential calculus, we used to call f’, the derivative of the function f. Here, in integral calculus, we call f as the anti-derivative or primitive of the function f’. And the process of finding the anti-derivatives is known as anti-differentiation or integration. As the name suggests, it is the inverse of finding differentiation.

It may seem strange that there exist an infinite number of anti-derivatives for a function f. Taking an example will clarify it. Let us take f’ (x) = \(3x^2\)

The integration of a function \(f(x)\)

\(∫f(x)~ dx\)

where R.H.S. of the equation means integral of \(f(x)\)

\(F(x)\)

\(f(x)\)

\(dx\)

\(C\)

\(x\)

Just like we had differentiation formulas, we have integration formulas as well. Let us go ahead and look at some examples.

Example: Find the Integral of the given functions

1. f(x) = \(\sqrt{x}\)

\(F(x)\)

We know that \(∫x^n~ dx\)

So, \(F(x)\)

2. a(n) = \(cos^{2n}\)

\(A(n)\)

We know that \(∫~f(kx)~dx\)

So, \(∫~cos~2n~ dn\)

\(∫~dn\)

\( A(n)\)

There are two broad parts of integral calculus which mainly serve the two purposes discussed in the beginning. They are indefinite integration and other is definite integration.

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