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^{2 }+ 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)].
∴ F(x) → INTEGRAL
∴ f(x)→ DERIVATIVE
Why do we need the constant of integration ‘C’ ? Consider the following examples –
x^{2}, x^{2} + 10, x^{2} -5.
If we differentiate these functions we get ,
d/dx (x^{2} ) = 2x.
d/dx(x^{2} + 10) = 2x+0 = 2x (as derivative of a constant is zero)
d/dx(x^{2} -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} |
[(x^{n+1})/n+1] + C , n≠ – 1 |
1/x |
ln |x| +C |
e^{x} |
e^{x}+ C |
e^{-x} |
-e^{-x }+C |
e^{kx} |
(1/k)e^{kx} +C |
a^{x} |
a^{x}/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 |