Integral Calculus

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.

Integral - Calculus

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) = \(3x^2\). By hit and trial, we can find out that its anti-derivative is \(F(x)\) = \(x^3\). This is because if you differentiate \(F\) with respect to \(x\), you will get \(3x^2\). There is only one function that we got as the anti-derivative of f. Let us now differentiate \(G(x)\) = \(x^3 ~+ ~9\) with respect to \(x\). Again we would get the same derivative i.e. \(f\). This gives us an important insight. Since the differentiation of all the constants is zero, we can write any constant with \(x^3\) and the derivative would still be equal to \(f\). So, there are infinite constants which can be substituted for \(C\) in the equation \(F(x)\) = \(x^3~ +~ C\). And hence, there are infinite functions whose derivative is equal to \(f\). And hence, there are infinite functions whose derivative is equal to \(3x^2\). \(C\) is called an arbitrary constant. It is sometimes also referred as the constant of integration.

The integration of a function \(f(x)\) is given by \(F(x)\) and it is represented by:

\(∫f(x)~ dx\) = \( F(x) + C \)

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

\(F(x)\) is called anti-derivative or primitive.

\(f(x)\) is called the integrand.

\(dx\) is called the integrating agent.

\(C\) is called constant of integration.

\(x\) is the variable of integration.

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) = \(\sqrt{x}\)

\(F(x)\) = \(∫f(x) ~dx\) = \(∫√x ~dx\) = \(∫x^{\frac{1}{2}}~ dx\)

We know that \(∫x^n~ dx\) = \(\frac{x^{n+1}}{n+1} ~+ ~C\), when n ≠ -1

So, \(F(x)\) = \(\frac{x^\frac{1}{2}~+~1}{\frac{1}{2}~+~1}\) = \(\frac{2}{3}~ x^{\frac{3}{2}}~ + ~C\)

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

\(A(n)\) = \(∫~cos^2n ~dn\)= \(\frac{1}{2} ∫~ 2 cos^2~n ~dn\) = \(\frac{1}{2} ∫~(cos~2n~-~1) dn\) = \(\frac{1}{2}[∫cos~2n ~dn~-~∫~1.dn]\)

We know that \(∫~f(kx)~dx\) = \(\frac{F(x)}{k}~ + ~ C\) (where \(k\) is some constant) and \(∫~cos~x ~dx\) = sin x + C

So, \(∫~cos~2n~ dn\) = \(\frac{sin~2n}{2} ~+ ~C_1\)

\(∫~dn\) = \( n~ + ~C_2\)

\( A(n)\) = \(\frac{1}{2}[\frac{sin~2n}{2} ~+~ C_1 ~- ~(n ~+ ~C_2)]\) = \(\frac{1}{2}[\frac{sin~2n}{2} ~- ~n]~ + ~C\), where \(C\) = \(\frac{C_1~-~C_2}{2}\)<

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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