14. ∫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
15. If ∫ f(x) dx = F(x), then ∫ f(ax ± b)dx = F(ax ± b).
3.
Theorems of Integration
16. 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]
22. 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
25. The integral is equal to
(where c is a constant of integration)
[JEE Main 2021]
A.
B.
C.
D.
26.
27. is equal to : (where C is a constant of integration)
A.
B.
C.
D.
[JEE Main 2021]
The integral
28.
29. If
and f(1) = 1/k, then the value of K is [JEE Main 2021]
30.
31. 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]
32. 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]
33.
34. 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]
35. The integral is equal to:
(where C is a constant of integration)
A.
B.
C.
D.
[JEE Main 2021]
36.
37. The integral is equal to :
(where C is a constant of integration)
A.
B.
C.
D.
[JEE Main
2020]
38.
39. 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]
68. 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]
69.
70. The integral is equal to
(where C is a constant of integration) :
A.
B.
C.
D.
[JEE Main
2020]
71.
72. 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
106. Type - 4
Divide Nr and Dr by x2 and take suitable substitution
Integrals of Trigonometric Functions:
107.
108.
109.
110.
111. Type - 5
(a) Substitute sin x = t, is n is odd;
(b) Substitute cos x = t, is m is odd
Approach:
Integrals of Trigonometric Functions:
112.
113.
114. Type - 5
(c) Substitute tan x = t, if m + n is a negative even integer.
Approach:
Integrals of Trigonometric Functions:
115. Type - 5
Approach:
(d) If m and n are rational numbers and
is a negative integer, then cot x = t or tan x = t
Integrals of Trigonometric Functions:
116.
117.
118. -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.
119.
120. Type - 6 sin x ± cos x = t
Integrals of Trigonometric Functions:
121.
122.
123. 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]
133. Type - 1
Type - 2
Type - 3
Type - 4
Integrals of Irrational Algebraic Function
134. Type - 1
Put px + q = t2
Integrals of Irrational Algebraic Function
135.
136. Type - 2 Put px + q = t2
Integrals of Irrational Algebraic Function
137.
138.
139. Type - 3
Put ax + b = 1/t
Integrals of Irrational Algebraic Function
140.
141.
142. 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
Integrals of Irrational Algebraic Function
143. Case - 2 If ax2 + bx + c is a perfect square say (lx + m)2 then
put lx + m = 1/t
Type - 4
Integrals of Irrational Algebraic Function
144. Case - 3 If b = 0; q = 0 e.g. then put x = 1/t
Type - 4
Integrals of Irrational Algebraic Function
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