11.Chemistry & Mathematics VII-Calculus(4)-Integration(1).

Today I begin writing about the second part of calculus – INTEGRATION.In the earlier posts we discussed how differentiation helps us to divide our data into small components.This procedure helps us to get our results with exactitude.But it is also mandatory to collect these small pieces later and bring them together to get the final result! Integration helps us to do just this !

What is Integration?

Integration is the act of bringing together smaller components into a single system.

In our study Integration is basically reverse of differentiation.

In differentiation ⇒ we break a system into smaller parts.
In Integration       ⇒ we bring the small parts together and find change over that time.

So, if the slope [dy/dx] of a function is known,the function itself can be found out.

Notation for Integration– 

‘∫’ is the symbol for Integration. It is called the integral sign.It is an elongated ‘S’ standing for ‘SUM’.(In old German and English ‘S’ was written using this elongated shape.)
∴ ∫y.dx,     y → function.

e.g.  ∫2x.dx = x+ C. In this expression –

2x  ⇒ Integrand/function being integrated.
dx ⇒  This term tells us that ‘x’ is the variable w.r.t which we are integrating the function.
C   ⇒  Arbitary constant of Integration.

Other notation is F(x) = ∫f(x).dx    [ Capital F(x) means integral of f(x)].


Why do we need the constant of integration ‘C’ ? Consider the following examples –

x2,  x2 + 10,  x2 -5.

If we differentiate these functions we get ,

d/dx (x2 ) = 2x.

d/dx(x2 + 10) = 2x+0 = 2x (as derivative of a constant is zero)

d/dx(x2 -5) = 2x -0 = 2x.

So, when we reverse this i.e we take integral of 2x, we can’t with surety say what the constant is(it could be any number 10, -5 etc). So, while integrating it is mandatory to introduce ‘C’ for indefinite integrals.

The following table gives the function and its integral –

Function f(x)

Indefinite Integral f(x).dx

constant, k

k x + C

x n

[(xn+1)/n+1] + C , n≠ – 1


ln |x| +C


ex+ C


-e-x +C


(1/k)ekx +C


ax/ln(a) +C

∫cf(x) dx,c= constant

c ∫f(x) dx

∫(f + g) dx

∫f dx + ∫g dx

∫(f – g) dx

∫f dx – ∫g dx

These rules will be useful while using integration in our further study. In the next post we will talk more about Integration.Till then ,
Be a perpetual student of life and keep learning…
Good day !
References and Further Reading –

Leave a Reply

Fill in your details below or click an icon to log in:

WordPress.com Logo

You are commenting using your WordPress.com account. Log Out / Change )

Twitter picture

You are commenting using your Twitter account. Log Out / Change )

Facebook photo

You are commenting using your Facebook account. Log Out / Change )

Google+ photo

You are commenting using your Google+ account. Log Out / Change )

Connecting to %s