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100 Integrals problem to practice
Preprint · January 2024
DOI: 10.13140/RG.2.2.25177.29283
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1
Z ∞
0.5⌊x⌋ dx = 2 => solution
0
2
Z π
(sin x + cos x)
ln 3
dx =
=> solution
(9 + 16 sin 2x)
20
4
0
3
(x + 1)
dx = ln (x + ln x) + c => solution
x(x + ln x)
Z
4
5
Z 1
(
Z 1
ln(x + 1)
π ln (2)
dx =
=> solution
2
x +1
8
0
√
√
2020
1 − x2020 −
1 − x2022 )dx = 0 => solution
2022
0
6
Z 1
xx dx =
0
7
Z π/3
nn
n=1
Z 1
x−x dx =
0
8
∞
X
(−1)n−1
(sin xcos x
sin x
=> solution
∞
X
1
=> solution
n
n=1 n
− cos xsin x
cos x
)dx = 0 => solution
π/6
9
Z π
(sin x+2 sin (2x)+3 sin (3x)+4 sin (4x)+5 sin (5x))2 dx = 55π => solution
−π
1
A surprisingly easy Geometric Integral
King comes to your help
3
Is MIT integration Bee this easy?
4
The Mysterious Integral
5
The Quarrelsome Integral
6
Sophomore’s Dream-i
7
Sophomore’s Dream-ii
8
The trigonometric towers integral
9
The Trigonometric BUS Integral
2
1
10
Z ∞
1
0
1 + x + x2 + x3 + x4 + x5
11
Z
tanh2 (x)dx = x − tanh x => solution
12
13
ln (1 + x)
π2
dx =
=> solution
x
12
0
Z 1
Z 1Y
∞ 0 k=0
14
15
1
1
dx =
=> solution
k
2
2
1+x
x
1 √
(x3 cos ( ) + ) 4 − x2 dx = π => solution
2
2
−2
Z 2
Z π/2
0
16
π
dx = √ => solution
3 3
√
2 3π
dx =
=> solution
(sin (x) + cos (x))2
9
q
3
tan (x)
Z 1
1
2
1
x ln (x)
dx =
ψ1 ( ) − ψ1 ( )
4
2
36
3
3
0 x +x +1
17
Z π/2
ln (sin (y))dy = − ln (2)
0
18
Z ∞
0
=> solution
π
=> solution
2
sin (x)
π
dx =
=> solution
x
2
10
Bro, Are you joking?
How easy is MIT Integration Bee?
12
Integrating using series in MIT Integration Bee
13
This is the most easiest difficult question in MIT Integration Bee
14
Chinese University Wifi Password
15
Beta Gamma Function in MIT Integration Bee
16
Stepping on some hard integrals
R π/2
17
proof of why 0 ln (sin (y))dy = − ln (2) π2
18
proof of Dirichlet Integral
11
2
19
ln (2 sin (x)) =
∞
X
n=1
20
ln (2 cos (x)) =
∞
X
−
cos (2nx)
=> solution
n
(−1)n+1
n=1
21
Z πq
2
cos (2nx)
=> solution
n
tan (x)dx => solution
0
22
Z ∞
0
sin (x)
π
dx =
=> solution
x
2
sin (xn )
dx => solution
xn
0
√
Z ∞
π
2
24
sin (x )dx = √ => solution
0
2 2
√
Z ∞
π
2
25
cos (x )dx = √ => solution
0
2 2
23
26
Z ∞
Z ∞
sin (xn )dx => solution
0
Z ∞
cos (xn )dx => solution
√
Z ∞
π
2
28
sin (x )dx = √ => solution
0
2 2
27
0
19
Two wonderful Series from Dr. Peyam-i
Two wonderful Series from Dr. Peyam-ii
21
Best application of Beta Gamma Function
22
Proof of Dirichlet Integral
23
Proof of Generalized Dirichlet Integral
24
Easiest way to prove Fresnel Integrals using Beta Gamma Functions
25
Easiest way to prove Fresnel Integrals using Beta Gamma Functions
26
Generalized Fresnel Integral-i
27
Generalized Fresnel Integral-i
28
Solving Fresnel Integral from Laplace Transform
20
3
29
1 + x2
dx => solution
0 1 + x4
Z 1
30
31
Z 1
arctan x
dx => solution
x
0
32
Z π
2
0
33
34
=> solution
x
dx => solution
sin (x)
log (−2023) => solution
Z 2023
log (−x)dx => solution
1
35
Z 1
x
xx
xx
xx
..
..
.
..
x.
dx = Diverges => solution
0
36
Z ∞
Γ(1 + ix)Γ(1 − ix)dx =
−∞
37
Z e
π
=> solution
2
W (z)dz => solution
0
29
Solving 20 integrals by Feynman Technique
Another easy integral from MIT Integration Bee
31
The Catalan Integral
32
Not the Dirichlet Integral
33
Shorts: What is log(-2023) ?
34
Let’s go to Complex World
35
The tower of x integral
36
My take on your ”A satisfying gamma function integral @maths505”
37
Integrating Lambert W function
30
4
where W(z) is Lambert W function
38
∞
X
(−n)(n−1) xn
W (z) =
n!
n=1
39 1
.
..
1.
1 e
1
1 e
1 e
1 e e
e
= Ω => solution
e
40
=> solution
xe
π
dx =
=> solution
2(e+1)
1+x
2(e + 1)
Z ∞
0
41
Z π
4
log (2 cos (x))dx => solution
0
42
Z 1Z 1
dxdy
=> solution
0 1 + x2 y 2
0
43
Z 3Z 9
x2
0
44
3
x3 ey dydx => solution
Z 8Z 2 √
0
√
3
45
x4 + 1dxdy => solution
y
dy
y dx = ey => solution
38
Deriving the series of Lambert W function
The Tower of 1/e
40
Freaking Irrational Integral
41
Integration using Fourier Series
42
Most satisfying Double Integral
43
I can solve the impossible Integral -i
44
I can solve the impossible Integral - ii
45
Is that even possible?
39
5
46
Z
((1−x)3 +(x−x2 )3 +(x2 −1)3 −3(1−x)(x−x2 )(x2 −1))dx = 0 => solution
47
Z π
2
0
ln (sec (x))
ζ(2)
dx =
=> solution
tan (x)
4
48
∞
X
1
n=0 n!
49
√
xy x
√ dxdy => solution
√
0 x y+y x
Z 1Z 1
0
50
∞
X
Hn
n
n=1 2
51
= e => solution
= ln(4) => solution
Z ∞ −t
e − e−tx
t
0
dt = ln (x) => solution
Words of an adolescent =>solution
52
Z ∞
e−t tx−1 dx = Γ(x) => solution
0
53
Z ∞
tm−1 (1 − t)n−1 dt = β(m, n) => solution
0
46
MIT Integration Bee Qualifier Exam P10
Dear @Maths505, here’s my approach
48
Proving using Beta Gamma Function
You can not have a more difficult proof than this
49
Using symmetricity in Integrals
50
A standard technique for such problems: use generating function for harmonic number
51
This is the best use of Feynman’s Method
52
Origin of Gamma Function
53
Origin of Beta Function
47
6
54
55
Z π
2
− π2
56
=> solution
1
dx => solution
1 + ex cos (x)
Z π
1
=> solutiondx
− π4 (1 + ex cos x )(sin4 x + cos4 x)
4
57
√ √2
2
√
2
√
2
√
2
√
2
√
2
√
2
√
2
√
2
√ . ...
2
=> solution
n
58
59
π2
Vn = n rn => solution
( 2 )!
∞
X
Vn = eπ => solution
n=2k
where Vn is volume of n dimensional Sphere
60
61
Z π
2
=> solution
tan i x dx => solution
0
54
Solving the easiest integral using hardest technique i.e. Ramanujan’s Master Theorem
You cannot get a more easier integral than this in MIT Integration Bee
56
The Legend of JEE Mains solved by only 5 percent students
57
This is the best use of Lambert W function
58
Deriving the formula for volume of n dimensional sphere
59
sum of the volumes of all n-dimensional spheres
60
sum of the volume of even dimensional spheres
61
A Complex triggy boi
55
7
Z ∞
cos (x)
dx => solution
−∞ x2 + 1
62
63
γ=
∞
X
(−1)m
m=2
Z ∞
64
1
65
Z 2
ζ(m)
=> solution
m
dx
√
=> solution
x x4 − 1
1
1
(x − 1) 2 (2 − x) 2 dx => solution
1
66
−1
√
limπ (1 + sin(x) − cos(x))tan(2x) = e 2 => solution
x→ 4
67
1 − x cot x
=> solution
x→0
x2
lim
68
69
70 1
a
Z ∞
sin(x)
dx => solution
x
−∞
1 − cos x
dx => solution
x2
−∞
Z ∞
+
1 1
1
+ =
=> solution
b x
a+b+x
62
Using Laplace Transform to solve for an absolutely gorgeous result
Short Animation Proof of this absolutely gorgeous result
64
MIN Integration Bee 2010 Qualifier Problem 8
65
MIT Integration Bee 2010 Qualifier Problem 25
66
An awesome limit problem
67
How high school student vs University student solve this limit?
68
A single liner solution using Maz Identity
69
How Undergrad. Vs Grad solve this integral?
70
Unseemingly hard Quadratic equation
63
8
71 d
1
−1
dx−1
72
(x)
∞
X
1
n=1
n2 − x2
73
1
dx 2
=
Z ∞
0
75
Z ∞
0
76
1
dx 2
(1) => solution
1
π cot(πx)
+ 2 => solution
−2x
2x
π 2n
=> solution
xp−1
dx = Γ(p)ζ(p) => solution
ex − 1
xp−1
dx = Γ(p)η(p) => solution
ex + 1
η(s) = (1 −
77
78
(x)
d2
∞
X
ζ(2n)
n=1
74
1
d2
2
ζ(s) => solution
2s
Z ∞
sin(x)
0
x2
Γ(x) =
3
Z ∞
dx => solution
e−t tx−1 dt => solution
0
71
WTF are these things?)
How come we have cot here?
73
Stanford Mathematics Tournament
74 p−1 x
x /e − 1 // Product of Eulers Gamma and Reimann zeta function interms of Bose
integral
75 p−1 x
x /e + 1 // Product of Eulers Gamma and Dirichlet eta function
76
Relation between Dirichlet Eta and Reimann Zeta Function
77
MIT Integration Bee: This is the best application of MAZ Identity
78
Euler Representation of Gamma Function
72
9
79
80
81
Γ(s) = n→∞
lim
n
k
ns Y
=> solution
s k=1 s + k
∞
Y
x
1
x
= xeγx
(1 + )e− n => solution
Γ(x)
n
n=1
ψ(x + 1) = −γ +
82
83
∞
X
1
k=1 k
Z ∞
0
85
1
=> solution
k+x
ψ(x + 1) = −γ + Hn => solution
ψ(x + 1) = −γ +
84
−
1 − xn
dx => solution
1−x
ψ(1 − n) − ψ(n) = π cot(nπ) => solution
1
2ψ(2m) = ψ(m) + ψ(m + ) + 2 ln(2) => solution
2
86
Z ∞
0
(1 − x sin
1
)dx => solution
x


1
1
1
87
 => solution
lim  √ 2
+√ 2
+ ..... + q
2
2
n→∞
2
2
n −0
n −1
n − (n − 1)
79
Gauss Representation of Gamma Function
Weierstrass Representation of Gamma Function
81
Infinite Sum Representation for Digamma Function
82
This is the most beautiful equation in mathematics, Deriving from Scratch
83
Integral Representation for Digamma Function
84
Reflection formula for Digamma Function
85
Duplication formula for Digamma Function
86
The classic Problem from MIT Integration BEE
87
Harvard MIT Maths Tournament
80
10
√
x
88
lim
x→0
Z ∞
=> solution
∞
X
1
1
=> solution
2
2
2
2
n=1 n + x n=1 n − x
∞
X
91
92
!csc x
n
1 X
1
√
lim √
=> solution
n→∞
n k=1 n + k
89
90
1+x
e
1
=> solution
−s
prime 1 − p
Y
ζ(s) =
f (s)g(s)ds =
Z ∞
L{f }(t) L−1 {g}(t)dt => solution
0
0
93
∞
X
f (n) =
0
n=1
94
95
Z ∞
L−1 {f }(t)
dt => solution
et − 1
∞
X
4n − 3
9
=
=> solution
2
4
n=2 n(n − 1)
Z ln(2) x
e − e2x + e3x − e4x
0
1 + ex + e2x + e3x
88
dx => solution
A high school limit problem from IIT JEE
Limit involving Reimann Sum
90
Two important infinite sums
91
Trivial Proof of Euler’s Prime Product Formula // Relation between Reimann Zeta
Function and Prime Numbers
92
Two amazing theorems of MAZ -I
93
Two amazing theorems of MAZ -II
94
MAZ theorem helps me solve this infinite sum
95
Monstrous JEE Advanced Integral
89
11
96
Z ∞
0
sin(x)
π coth(π) − 1
dx =
=> solution
x
e −1
2
Z ∞
97
0
98
x2 − 1
dx => solution
x4 ln(x)
Z ∞
1
99
Z ∞
0
100
101
Z ∞
1
103
sin3 (x)
dx => solution
x2
sin(x)
= 1 => solution
x→0
x
lim
∞
X
(−1)n ζ(n)
n=2
102
sin(ax)
dx => solution
xn
2n
=> solution
{x}
1 ζ(3)
dx = −
=> solution
4
x
2
3
Z 1
2
1
4
104
1
⌊log⌊ ⌋⌋dx => solution
x
x7 − 1
dx => solution
0 log(x)
Z 1
96
MAZ theorem helps me solve this integral
Smashing an improper integral using MAZ Identity
98
MAZ Identity speed rockets the integral
99
MAZ Identity speed rockets the integral
100
5 Unusual ways to prove this limit
101
DIGamma Function helps me solve this infinite sum
102
Integration of Fraction Part for IIT JEE
103
A tricky GIF Integral from MIT Integration Bee
104
MIT Integration BEE Problem that needed MAZ Identity
97
12
105
106
Z 2022
L{ln(x)} => solution
x2 − ⌊x⌋⌈x⌉dx =
0
107
Z 1 X
∞
2
0
108
Z ∞
0
109
Z ∞
0
Cn xn dx => solution
n=0
π
π
sin(x)
dx = tanh( ) => solution
sinh(x)
2
2
Z 1√
Z 2√
0
1
1 − x2 dx
110
111
n+3
2022
=> solution
3
x2 − 1dx => solution
∞
X
1
1
−
=> solution
n+3
n=0 n + 2
a
f (ax) − f (bx)
dx = (f (∞) − f (0)) ln( ) => solution
x
b
112
∞
X
(n!)2
2π
= √ => solution
3 3
n=0 (2n + 1)!
i
113 d
dxi
(xi ) = i! => solution
105
Laplace Transform of ln(x)
Marriage of floor and ceiling function
107
A good problem from MIT Integration Bee
108
An Ridiculously Awesome Integral from Ramanujan’s land (India)
109
Solving Integrals Geometrically
110
An Introduction to extremely difficult way to do a simple telescoping sum
111
Frullani’s Integral
112
This ridiculously interesting sum is solved by Beta Function
113
Imaginary Derivative of imaginary number. wow
106
13
114
n!
nn
lim
n→∞
115 d
s n→∞
117
118
119
1
=> solution
e
(xπ ) = π! => solution
Z
f (x)dx ,
lim
n→∞
n!
nn
lim
n!
nn
n→∞
120
a
b
dx
=
1
2
3
n
Γ
Γ
.......Γ
n
n
n
n
lim n Γ
121 d
n
π
dxπ
116
!1
a
b
f (x),
Z
=> solution
δ
f (x) => solution
δx
!1
=
1
=> solution
e
=
1
=> solution
e
n
!1
n
∞! => solution
f (x)dx ,
δ
f (x) => solution
δx
114
A brilliant limit from Stanford Maths Tournament
Differentiation IIT JEE Maths — πth derivative — Application of Derivative
116
The is the most beautiful problem I ever solved
117
Product Integral and Product Derivative
118
Using the powerful stirling approximation for this IIT limit
119
Proving this IIT Limit using product integral
120
Infinity factorial, Happy Birthday Bishnu
121
500 sub special:: Inventing Math: Fractional Derivative, Product Integral and Product
Derivative
115
14
122
Z π
2
sin(x)dx => solution
0
123
Z 1
e−x ln2 (x)dx =
0
124
3ζ(3) π 2 π ln(2)
tan (xyz)dxdydz = −
−
+ −
=> solution
32
48 4
2
0
Z 1Z 1Z 1
0
0
125
π2
+ γ 2 => solution
6
−1
Z 1Z 1Z 1
0
126
0
f (xyz)dxdydz =
0
Z 1Z 1
0
f (xy)dxdy = −
1Z 1 2
ln (x)f (x)dx => solution
2 0
Z 1
ln(x)f (x)dx => solution
0
0
127
Z 1Z 1Z 1
128
0
0
10
X
10
e−xyz dxdydz => solution
0
Ck k 2 = 28160 => solution
k=0
129
Suggest your favorite integral
in the comment for upcoming Video =>solution
130
n + n2 + n3 + ...... + nn
1
=1−
=> solution
n→∞ 1n + 2n + 3n + ...... + nn
e
lim
122
Impossible seeming Integrals
Integral with two important constants
124
Ridiculously Awesome Impossible Integral
125
Ridiculously Awesome Impossible Integral
126
Ridiculously Awesome Impossible Integral
127
Ridiculously Awesome Integral
128
A sum from World International Mathematics Olympiad Final 2019
129
Suggest
130
The nightmare limit Problem
123
15
131
√
n
lim π(n)
n→∞
1
132
n − 1 = 1 => solution
2
3
n
⌊e n ⌋ + ⌊e n ⌋ + ⌊e n ⌋ + .... + ⌊e n ⌋
lim
=> solution
n→∞
n
Z ∞
133
⌊x⌋e1−⌊x⌋ dx => solution
0
134
F (n) =
x2n−2
dx, F (5) =? => solution
(x4 − x2 + 1)n
Z ∞
0
135
lim
n→∞
136
0
R .∞
R . xdx
R 11
xdx
137
√
Z 1Z 1
0
R 1
Z R 1
(2n)!.(2n + 1)!
=> solution
(n! .2n )4
x+
√
y
q√
xy(1 − xy)
dxdy => solution
xdx
xdx
xdx
1
xdx = 2 +
1
138
2023
X
n=0
5n +
1
2 + 2+
=> solution
1
1
2+ 2+.1
.∞
1
√
=> solution
52023
131
Limit involving Prime counting Function
JEE Advanced Limit (Model Question)
133
Easy Integral by Himanshu
134
Hard Integral by Himanshu
135
This is the best use of stirling’s approximation
136
This is the best use of digamma function
137
Surprise!!!
138
Sum involving King’s Rule
132
16
Z ∞
x−1
sin(x)
π Z 1 sin(ln(x))
π
dx = ln(2),
= ,
= ,
x
2 0
ln(x)
4
0 ln(x)
0
√
q
Z
Z ∞ −x2
∞
√
e
sin(x2 )
π
π − 25
2
),
e−x cos(5x)dx =
e 4 => solution
dx
=
π
2
sin(
2
x
8 0
2
0
139
Z 1
Z
140
141
W (x)dx => solution
∞
X
1
=> solution
n=0 (4n)!
0.05 0.05 (2×17+0.05×202 −2x)
17 + 0.05 × 202 − x
√
e 2
× erf c
2
2 × 20
min = f (x) =
142
Z ∞
!!
2
e−x dx => solution
0
143
144
dx
147
Z π
2
Z q
ln(sin(x))dx
Z π
2
π x
− | 2) + c => solution
4
2
ln(cos(x))dx
0
Z π
4
0
W (x) => solution
sin(x)dx = −2E(
0
148
sin(sin(x))
Adif f icultintegralproblemmadeeasy, shorts
145 d
146
Z
ln(sin(x))dx
Z π
2
ln(tan(x))dx => solution
0
Z π
4
ln(cos(x))dx
0
Z π
4
ln(tan(x))dx => solution
0
139
Destroying five harsh integrals using Feynman’s Technique
Integral of Lambert W function
141
Can you solve this sum?
142
6 proofs of Gaussian Integral
143
Horse shoe Integral
144
MIT would not want to listen this hack about MIT Integration BEE
145
Differentiation of Lambert W function
146
Elliptic Integral of the second kind
147
A nice family of Integrals
148
A nice family of Integrals
140
17
×
3030
0.0153
149
Z π
4
ln(1 + tan(x))
0
150
Z ∞
0
Z π
4
ln(1 − tan(x)) => solution
0
Z 1 2
Z 1 2
Z ∞ 2
ln2 (x)
ln (x)
ln (x)
ln (x)
dx
dx
dx
dx => solution
1 − x2
1 + x2
0 1 − x2
0 1 + x2
0
151
1. Proof of Lhopital’s rule
2. Fundamental theorem of calculus in a visual way
3. Calculation of Escape Velocity by Newton in 17th Century
4. Introduction to Epsilon-Delta Definition
5. Zeno’s paradox in limits
6. What does it mean to be undefined at a point but have limiting value at
a point
7. Different notation for differentiation of Newton and Leibniz
8. Rigorous proof of Euler’s Identity from level 0 9. Applications of
Differential equation: NEwton’s Law of Cooling 10. When to swap the sum
and integrals 11. Why does the nth root test work?
152
1. Fractional root of a Matrix, exponential of a matrix, logarithm of a
matrix
2. Contour Integration 3. Usage of Epsilon-Delta Definition and when does
it fail ? 4. Deriving Gamma’(1)= - gamma and Gamma”(1) =
gamma2 + zeta(2)anddigamma(1/2) = −gamma +
2ln(2)5.F indingthevalueof ReimannZetaof 2f romLevel0throughDigammaF unction
153
Z q
tan(x)dx
Z q
cot(x)dx => solution
149
One of them is easy and other is hard
A happy get-together of integrals
151
Some cool concepts to explain
152
Some videos on my Checklist
153
A story of two brothers
150
18
154
155
sin(z) = 2 => solution
Z ∞
x cos(x)
dx => solution
ex − 1
0
156
Z π
πZπ
f (sin(x))dx => solution
2 0
xf (sin(x))dx =
0
157
Z
f −1 (x)dx = xf −1 (x) − F (f −1 (x)) + c => solution
158
Z b
f (x)dx =
a
dx
Z b
f (a + b − x)dx => solution
a
159 d
160
Z b
(f −1 (x)) =
f (x)dx +
Z f (b)
1
f ′ (f −1 (x))
=> solution
f −1 (x)dx = bf (b) − af (a) => solution
f (a)
a
161
n
(f (x)g(x)) =
n
X
n
Cr f r (x)g n−r (x) => solution
r=0
162 d
Z b
dy
a
!
f (x, y)dx =
Z b
∂
(f (x, y))dx => solution
a ∂y
154
This has a solution!!!
A Ridiculously Awesome integral
156
A nice Lemma for my nice viewers
157
Proof and Usage of Inverse Integration Technique
158
Proof (algebraic and geometric) and usage of King’s Rule
159
Proof and usage of inverse derivative technique
160
Proof(algebraic and geometric) and usage of definite inverse integration technique
161
Verification and usage of Leibniz Rule
162
Proof and application of Feynman’s Technique
155
19
163
164
Z a
F orf (x, y) = 0
odd(x)dx = 0
Z a
−a
even(x)dx = 2
Z b
f (x)dx = −
Z a
f (x)dx => Solution
b
Z
f (x)g(x)dx = f
even(x)dx => Solution
0
a
Z
Z a
−a
165
166
dy
fx
=−
=> solution
dx
fy
g −f ′
Z Z
g +f ′′
Z Z Z
g −f ′′′
Z Z Z Z
g +...... => Solution
√
π Γ( n2 )
=> Solution
sech (t)dt =
Γ( n+1
)
−∞
2
Z ∞
n
167 z
1 = 3 => Solution
168
Z π
2
0
169
x
dx => Solution
tan(x)
β(m, n) =
Γ(m)Γ(n)
=> Solution
Γ(m + n)
n
∞
Y
nz Y
k
Γ(x)Γ(y)
170
Γ(z) = n→∞
lim
|||
=
z k=1 z + k Γ(x + z)Γ(y − z) k=0
163
"
z
1+
x+k
z
1−
y+k
proof and application of complete differentiation using partial differentiation
proof (algebraic and geometric) and usage of odd/ even function
165
Proof and usage of reflection formula
166
Proof and Usage of DI(Differentiation Integration) Method
167
Everything is possible in the realm of complex numbers
168
Feynman’s Technique is never obvious
169
Proof of beta-gamma function using Laplace Tranform and convolution Integral
170
A simple problem involving Gauss Representation of Gamma Function
164
20
!#
=> Solution
171
√
Z ∞ −t
e cosh(a t)
√
0
172
173
t
dt => Solution
(2n)! √
1
π => solution
Γ(n + ) = n
2
4 n!
Γ(x)Γ(1 − x) =
174
π
=> Solution
sin(πx)
√
1
− ! = π => Solution
2
4.4 6.6 8.8
π
. . .... =
=> Solution
1.3 3.5 5.7 7.9
2
175 2.2
.
1
1
1
π2
176 1
+
+
+
+
.....
=
12 22 32 42
6
177 sin(πx)
πx
178
=
∞
Y
x2
1− 2
n
!
sin(x) +
q
n=1
Z π
2
ln
q
=> Solution
=> Solution
cos(x) dx
0
179
1
= −γ − 2 ln(2) => Solution
2
ψ
171
This is the best use of Legendre’s Duplication Formula
A common sense proof of Legendre’s Duplication formula
173
Proving the Euler’s Reflection using Sine Product Formula
174
Finding (-1/2)! without gaussian integral
175
Proving Wallis Product using Sine Product Formula
176
Finding Reimann zeta function of 2 using Sine product formula
177
Proving the Sine Product Formula using Digamma Function
178
A symmetric Integral
179
Finding the value of digamma(1/2)
172
21
180
Z ∞Z ∞
0
0
181
!
Z 1Z 1Z 1
1
1
ln
+
dxdydz => Solution
1 + xyz 1 − xyz
0
0
0
182
tan−1 (x2 ) tan−1 (y 4 )
dxdy => Solution
x2 y 3
Z ∞ Z ∞
−∞
−∞
183
1
1 + x2 + y 2
zn
Z 1
0
184
(1 − z)
1
2
!n
dxdy, nϵN, n > 1 => Solution
dz = 2.
(2n)!!
=> Solution
(2n + 1)!!
1
= −γ − 2 ln(2) => Solution
2
ψ
180
A simple problem for practice
A bonus assignment problem from my mentor
182
This is the best use of polar coordinates
183
When can double factorial be helpful?
184
Finding digamma (1/2) without using Legendre’s Duplication Formula
181
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