Hg
Name: Class 12 STICK TO YOUR WALL IN STUDY AREA
___________________________________________________________________________________________________________________________________
___________________________________________________________________________________________________________________________________
The Hg Classes (8
th
to 12
th
) By: Er Hershit Goyal (B.Tech. IIT BHU), 134-SF, Woodstock Floors, Nirvana Country, Sector 50, GURUGRAM +91 9599697178.
fb.me/thehgclasses linkedin.com/company/the-hg-classes instagram.com/the_hg_classes g.page/the-hg-classes-gurugram thehgclasses.co.in
INTEGRATION
• Integration is the inverse process of differentiation i.e., the process of finding the function whose
differential is given.
• Given d/dx F(x) = f(x), then
= F(x) + C. These integrals are called Indefinite Integrals or general
integrals. All these integrals differ by a constant.
o f(x) is called Integrand.
o F(x) is called Integral.
o C is the constant of Integration.
o x is the variable of integration
• Geometrically, all these integrals represent a family of curves, each of which is obtained by translating
one of the curves vertically along the y – axis.
Standard Integral Formulae (obtained by corresponding derivative formulae):
1.
= C
2.
= x + C
3.
=
, n ≠ -1
4.
= - cos x + C
5.
= sin x + C
6.
= tan x + C
7.
= - cot x + C
8.
= sec x + C
9.
= - cosec x + C
10.
dx = sin
-1
x + C
11.
dx = - cos
-1
x + C
12.
dx = tan
-1
x + C
13.
dx = - cot
-1
x + C
14.
dx = sec
-1
x + C
15.
dx = - cosec
-1
x + C
16.
=
+ C
17.
=
+ C
18.
dx = + C
Integration by Substitution: This technique is used when derivative of some part of integrand function is
also present in the Integrand, and then substituting that part with another variable.
19.
= ln |sec x| + C
20.
= ln |sin x| + C
21.
= ln |sec x + tan x| + C
22.
= ln |cosec x – cot x|+ C
Integration by Partial Fractions: This technique is used for rational functions P(x) / Q(x) where degree of
P(x) <= degree of Q(x). i.e., proper rational functions. By using partial fraction decomposition as per below
table we write the original function as a sum of simpler rational functions, and then carry out integration
by already known methods.
Form of the rational function
Corresponding Partial Fraction
1.
a and b are different
+
2.
+
3.
a, b and c are all different
+
+
Hg
Name: Class 12 STICK TO YOUR WALL IN STUDY AREA
___________________________________________________________________________________________________________________________________
___________________________________________________________________________________________________________________________________
The Hg Classes (8
th
to 12
th
) By: Er Hershit Goyal (B.Tech. IIT BHU), 134-SF, Woodstock Floors, Nirvana Country, Sector 50, GURUGRAM +91 9599697178.
fb.me/thehgclasses linkedin.com/company/the-hg-classes instagram.com/the_hg_classes g.page/the-hg-classes-gurugram thehgclasses.co.in
4.
a and b are different
+
+
5.
+
Some more standard Integrals (obtained by techniques of partial fractions and substitution):
23.
=
ln|
| + C
24.
=
ln|
| + C
25.
=
tan
-1
+ C
26.
= sin
-1
+ C
27.
= ln |x +
+ C
28.
= ln |x +
+ C
Integration by Parts: This technique is used for integrands which are product of two functions say f(x) *
g(x). g(x) is chosen to be that function whose integral we can calculate easily by known techniques, and f(x)
is chosen to be that function whose derivative can be calculated easily.
= f (x)
–
f’(x)
] dx
Integral of the form:
= e
x
f(x) + C
Some more standard Integrals:
29.
dx =
-
ln |x +
+ C
30.
dx =
+
ln |x +
+ C
31.
dx =
+
sin
-1
+ C
