Indefinite Integration One shot Revision

Indefinite Integration
One Shot

Nishant Vora
B.Tech - IIT Patna

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Integration is the reverse process of differentiation.
Indefinite Integration

1. ∫ xn .dx =
xn+1
n + 1
+ c, n ≠ -1
2.
Standard Integrals
3.
4.

5.
6.
Standard Integrals
7.
8.

9.
10.
Standard Integrals
11.
12.
13.

∫k f(x) dx = k∫f(x) dx, where k is any constant.
1.
If f1(x), f2(x), f3(x) ….. (finite in number) are functions of x, then
∫[f1(x) ± f2(x) ± f3(x) …..] dx = ∫f1(x) dx ± ∫f2(x) dx ± …...
2.
Theorems of Integration

If ∫ f(x) dx = F(x), then ∫ f(ax ± b)dx = F(ax ± b).
3.
Theorems of Integration

If
and f(0) = 0, then f(1) is equal to :
A.
B.
C.
D.
π + 1 / 4
1 / 4
π - 1 / 4
π + 2 / 4
[JEE Main
2020]

Substitution
By part (product rule)
Partial (fraction)
Miscellaneous
Techniques of Integration

i. ∫ tan x dx = ln (secx) + C OR - ln(cosx) + C
ii. ∫ cot x dx = ln (sinx) + C
iii. ∫ sec x dx = ln (secx + tanx) + C OR
iv. ∫ cosec x dx = ln (cosecx - cotx) + C OR
Substitution

i. ∫ tan x dx = ln (secx) + C
Substitution

iii. ∫ sec x dx
Substitution

The integral is equal to
(where c is a constant of integration)
[JEE Main 2021]
A.
B.
C.
D.

is equal to : (where C is a constant of integration)
A.
B.
C.
D.
[JEE Main 2021]
The integral

If
and f(1) = 1/k, then the value of K is [JEE Main 2021]

If
and f(0) = 0, then the value of f(1) is :
A.
B.
C.
D.
-1/2
-1/4
1/2
1/4
[JEE Main 2019]

If
a constant of integration, then λf(π/3) is equal to :
A.
B.
C.
D.
-9/8
2
9/8
-2
[JEE Main
2020]

If
Where C is a constant of integration, then the function f(x) is equal to :
A.
B.
C.
D.
3/x2
- 1/6x3
- 1/2x2
- 1/2x3
[JEE Main 2019]

The integral is equal to:
(where C is a constant of integration)
A.
B.
C.
D.
[JEE Main 2021]

The integral is equal to :
(where C is a constant of integration)
A.
B.
C.
D.
[JEE Main
2020]

If
where C is a constant of integration, then the ordered pair
(λ, f(θ)) is equal to :
A.
B.
C.
D.
(1, 1 - tanθ)
(-1, 1 - tanθ)
(-1, 1 + tanθ)
(1, 1 + tanθ)
[JEE Main
2020]

General
Substitution and
Loving Integrals

★
★
★
★ ; x2 = a2 cos 2θ
; x = a sin θ
; x = a tan θ
; x = a sec θ
General Substitutions

Let
dx (x ≥ 0). Then f(3) - f(1) is equal to :
A.
B.
C.
D.
[JEE Main
2020]

constant of integration, then :
[JEE Main 2019]
A.
B.
C.
D.

1.
2.
3.
4.
5.
Loving Integrals

★
For integration of type. and
make ax2 + bx + c as perfect square
For integration of type and
write px + q = 𝝺(2ax + b) + μ
★
Note

Example Evaluate ∫ x tan-1 x dx
Type 1 : Separate functions are visible

Example Evaluate ∫ lnx dx
Type 2 : Considering ‘1’ as a function

Example Evaluate ∫ ex cos x dx
Type 3 : Recurring By Parts

a.
b.
∫ ex (f(x) + f’(x)) dx = ex f(x) + C
∫ (f(x) + xf’(x)) dx = x f(x) + C
Type 4 : Special Cases

Example Evaluate
Type 4 : Special Cases

The integral equals :
A.
B.
C.
D.
e(4e + 1)
4e2 - 1
e (4e - 1)
e (2e - 1)
[JEE Main
2020]

The integral dx is equal to
A.
B.
C.
D.
[JEE Adv 2014]

where C is a constant of integration, then the ordered pair (A(x), B(x))
can be :
A.
B.
C.
D.
(x + 1, - √x)
(x + 1, √x)
(x - 1, - √x)
(x - 1, √x)
[JEE Main
2020]

The integral is equal to
(where C is a constant of integration) :
A.
B.
C.
D.
[JEE Main
2020]

If
where C is a constant of integration, then f (x) is equal to :
A.
B.
C.
D.
-2x3 - 1
-4x3 - 1
-2x3 + 1
4x3 + 1
[JEE Main 2019]
DIY

Integration
By Partial Fraction

If
where C is a constant of integration, then B(θ)/A can be :
A.
B.
C.
D.
[JEE Main 2020]

★
★
Partial Fraction can be applied only if
Deg (Nr) < Deg (Dr)
If Deg (Nr) ≥ Deg (Dr) then Long
Division
Remember:

★
Partial Fractions Involving Even Powers of x only:

Partial Fractions Involving Even Powers of x only:

Integrals of
Trigonometric
Functions

Type - 1
Multiply Nr and Dr by sec2x or cosec2x and proceed
Integrals of Trigonometric Functions:

Type - 2
Convert sinx and cosx into their corresponding tangent
to half the angles and put

Type - 3

Evaluate the indefinite integral

Type - 4
Divide Nr and Dr by x2 and take suitable substitution

Type - 5
(a) Substitute sin x = t, is n is odd;
(b) Substitute cos x = t, is m is odd
Approach:

Type - 5
(c) Substitute tan x = t, if m + n is a negative even integer.
Approach:

Type - 5
Approach:
(d) If m and n are rational numbers and
is a negative integer, then cot x = t or tan x = t

-3 tan-1/3 x + C
-3 cot-1/3 x + C
-¾ tan-4/3 x + C
3 tan-1/3 x + C
The integral x dx is equal to
(Here c is a constant of integration)
[(JEE M 2019 - 9 April (M))
A.
B.
C.
D.

Type - 6 sin x ± cos x = t

If
where c is a constant of integration, then the ordered pair (a,b) is equal
to :
A.
B.
C.
D.
(1, -3)
(1, 3)
(-1, 3)
(3, 1)
[JEE Main 2021]

(ii)
(iii)
(i)
Integrals of √Quadratic

Integrals of
Irrational
Functions

Type - 1
Type - 2
Type - 3
Type - 4
Integrals of Irrational Algebraic Function

Type - 1
Put px + q = t2

Type - 2 Put px + q = t2

Type - 3
Put ax + b = 1/t

Type - 4
Case - 1 When (ax2 + bx + c) breaks up into two linear factors, e.g
Put x - 2 = 1/t Put x + 1 = 1/t

Case - 2 If ax2 + bx + c is a perfect square say (lx + m)2 then
put lx + m = 1/t
Type - 4

Case - 3 If b = 0; q = 0 e.g. then put x = 1/t
Type - 4

Integral Substitution
Px + q = t2
ax + b = 1/t
x = 1/t
Px + q = t2
Summary

1.
2.
3.
4.
5.
Practice Questions

Reduction formula of ∫secn x dx
Reduction Formula
Reduction formula of ∫cosecn x dx

If
such that I2 = αI1 then α equals to :
A.
B.
C.
D.
5049/5050
5050/5049
5050/5051
5051/5050
[JEE Main
2020]

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Indefinite Integration One shot Revision

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