Page 50 - INSPERIA-School Magazine TIGPSKON 2021.indd

Integral Calculus

Integral Calculus applications throughout the Maths field.
Integration forms one of the most important aspects of
calculus. Now, often the question comes that what is
“Integration”? Well, in simple terms, it can be said to be an
operator which is used to find out the area under the curve

in a graph.
The graph of this equation is a bell curve:

The birth of Integral Calculus took place in India but its
main contributions are done by Sir Isaac Newton.

The general form of a definite integration is:
GAMMA AND PI FUNCTIONS
; where ‘a’ and ‘b’ are called the upper and lower

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limits respectively and f(x) is a function in x. The factorial definition that we all know is barely restricted
Few Integral Formulae
1. x dx = x /(n+1) + C, where C is an arbitrary constant
n+1
n
and n ≠ 1.
2. ʃ x dx = ln |x| + C
-1
3. ʃ a dx = a /(ln a) + C
x
x
4. ʃ e dx = e /m + C within the whole numbers (since 0!=1). However, the basic
mx
mx

5. ʃ sinx dx = – cosx + C definition of factorial is after all given by two very interesting
6. ʃ cosx dx = sinx + C functions, namely, the Gamma and Pi functions.
7. ʃ tanx dx = ln |secx| + C
8. ʃ secx dx = ln |secx + tanx| + C This notation for the Gamma function was given by
9. ʃ cosecx cotx dx = – cosecx + C Legendre. Now, the Pi function [denoted by Π(z)] is almost
10. ʃ secx tanx dx = secx + C similar to that of the Gamma function. Just, in place of the
(z – 1), it will be only z.
FEYNMAN’S TECHNIQUE FOR FINDING Sinx/x
=
I(b) =
Using the integration by parts method after partially
differentiating the function and then re-integrating it, we
will be left with:

-1
I(b) = – tan b + π/2. However, our original Integral was
I(0). So, on substituting b = 0, we get the inal result of
π/2 as our answer.
Aditya Majumder
GAUSSIAN INTEGRAL Class XI
It is one of the handiest integral tools that has immense

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