Mathematics_Today_-_October_2016

* Alok Kumar is a winner of INDIAN NATIONAL MATHEMATICS OLYMPIAD (INMO-91).

He trains IIT and Olympiad aspirants.

Integration of a function is the reverse process of
differentiation. If f ′(x) is the derivative of a function
f(x), then the process of finding f(x) from f ′(x) is called
integration.

ALGEBRA OF INTEGRATION
(i) ∫∫( ()U x±= ±V x dx()) ∫U x dx() V x dx()
(ii) ∫∫Kf x dx()⋅ =K f x dx() ,

where K is a real number (constant).

SOME FUNDAMENTAL INTEGRALS

(i) xdxx

n

n cn

n

∫ ⋅ = + + ≠−

+ 1

1

, where 1

(ii)^10

x

∫ dx=+log | |e x c, wherex≠

(iii) ∫edx e cxx=+

(iv) adx a

a

x ca

x

∫ e

=+ >

log

, where 0

(v) ∫sinxdx=−cosx c+

(vi) ∫cosxdx=+sinx c

(vii) ∫sec^2 xdx=+tanx c

(viii)∫cosec^2 xdx=−cotx c+

(ix) ∫secxxdx xc⋅tan =+sec

(x) ∫cosecxxdx⋅cot =−cosecxc+

(xi)

dx

x

xc xc x

1

2 11 1

−

∫ =+sin−−or−cos +<,where| |

(xii)^1

1 2

11

+

∫ =+−−− +

x

dxtan xcor cot xc

(xiii)

1

(^21)
11
xx
dx xc xc
−
∫ =+sec−−or−cosec +
SOME STANDARD INTEGRALS USING THE
ABOVE RELATIONS ARE SHOWN BELOW
(i) ∫tanxdx=+=log | sec |x c −log | cos |x+c
(ii) ∫cotxdx=−log | cosec |x c+=log | sin |x c+
(iii) ∫secxdx=++log | secx tan |x c
(iv) ∫cosecxdx=log |cosecx−cot |x+c
=+log tanx c
2
SOME SPECIAL INTEGRALS
(i) dx
xaa
x
a
221 1 c

• ∫ =+tan−
  (ii) dx
  xa a
  xa
  xa
  221 c
  − 2
  = −


• 
  

  ∫ log +
  (iii)
  dx
  ax a
  ax
  ax
  221 c
  − 2
  = +
  −
  ∫ log +
  (iv) dx
  ax
  22 xxac^22
  ±
  ∫ =+±+log
  (v)
  dx
  ax
  x
  a
  22 1 cx a
  −
  = ⎛
  ⎝⎜
  ⎞
  ⎠⎟
  ∫ sin− +<, | | | |^
  
   
  This column is aimed at Class XII students so that they can prepare for competitive exams such as JEE Main/Advanced, etc. and
  be also in command of what is being covered in their school as part of NCERT syllabus. The problems here are a happy blend
  of the straight and the twisted, the simple and the difficult and the easy and the challenging.
  *ALOK KUMAR, B.Tech, IIT Kanpur