1. Find the value of ( ) ( )4
3
4
1 ΓΓ
8. Evaluate ( ) ( ) .73 4
17
0
4
1
dxxx −−∫
Prove that )1( +Γ n = n!
6. Write the formula for θθ
π
dn
∫
2
0
cos .
Define Beta and Gamma function.
8. Show that
2
2 1 2 1
0
1
sin ( )cos ( ) ( , )
2
m n
d m n
π
θ θ θ β− −
=∫ .
Define Beta function.
6. Compute .
)3()5(
)8(
ΓΓ
Γ
Write the formula for θθθ
π
dmn
cossin
2
0
∫ .
6. Evaluate θθ
π
d∫
2
0
tan
Evaluate ∫ ∫
2/
0
sin
0
π θ
θddrr
6. Evaluate ∫∫∫
1
0
2
0
3
0
dxdydzxyz
7. Prove that ∫ ∫ −=
a a
dxxafdxxf
0 0
)()(
8. Show that
2
4
3.
4
1
tan
2/
0
=∫ θθ
π
d

2. 1. Evaluate ∫
π
θθθ
2
43
o
dcossin
2. State any two properties of Beta function.
3. Define Beta and Gamma functions.
4.
5. 6. Prove that .
)1,(
),1(
n
m
nm
nm
=
+
+
β
β
6. Prove that Γ(α + 1) = α Γ α.
7.
8. 6. Define β(m, n) and prove β(m, n) = β(n, m).
What is the reduction formula for (n)?
6. Define Beta function.
Give two integers such that their Gamma values are equal.
6. Write ∫
2
0
22
sincos
π
θθθ dnm
in terms of Beta integral.
Define Gamma and Beta function.
6. Prove that .
8
3
0
4 2
π=∫
∞
−
dxex x

3. (a) Using Beta and Gamma functions evaluate ( )∫
−
−
1
1
2
,1 dxx
n
where
‘n’ is positive integer.
(b) Evaluate .
1
log
31
0
4
dx
x
x 











∫
(or)
18. (a) Prove that
( )
( )
.
,
,1
nm
m
nm
nm
+
=
+
β
β
(b) Evaluate ( )∫
1
0
3
.log dxxx
(a) Prove that ( ) ( )mmm m
,2
2
1, 12
ββ −
= .Hence find )2( mΓ .
(b) Show that nm
nm
n
nm
m
m
+
=
+
=
+ ),()1,()1( βββ
.

4. (a) Evaluate
1
0
(1 )m n p
x x dx−∫ in terms of gamma function and
hence find
1
0 1 n
dx
x−
∫ .
(b) Find the value of
1 1m n
x y dx dy− −
∫∫ over the positive
quadrant of the ellipse
2 2
2 2
1
x y
a b
+ = in terms of gamma
function.
Prove that )(
)()(
),(
nm
nm
nm
+Γ
ΓΓ
=β where m,n > 0.
(or)
16. Express ∫ −
1
0
)1( dxxx pnm
in terms of Gamma functions and
evaluate ∫ −
1
0
1035
)1( dxxx .
(a) Prove that ( ) ( )mmm m
,2
2
1, 12
ββ −
= .Hence find )2( mΓ .
(b) Show that nm
nm
n
nm
m
m
+
=
+
=
+ ),()1,()1( βββ
.
Prove that β (m,n) = )( nm
nm
+
1. (a) Prove that
( ) ( ) ( )
nm
n,m
m
n,1m
n
1n,m
+
β=
+
β=
+
β
(b) Show that .on,ndy
y
log
n
o
>Γ=




−
∫
11
1
(or)
(b) Express ∫
−
1
4
1o x
dx
in terms of Gamma function

5. Prove that β(m,n) = ∫
∞
+
−
+0
1
.
)1(
dx
x
x
nm
m
Hence deduce that β(m,n)
= ∫ +
−−
+
+
1
0
11
.
)1(
dx
x
xx
nm
nm
.
(or)
16. Express ∫ −
1
0
)1( dxxx mqp
in terms of Gamma functions.
Hence Evaluate dxxx∫ −
1
0
3
)1(
Using Beta and Gamma function, show that for any positive
integer m
(a) .
3)...32)(22(
2)...42)(22(
)(sin
2
0
12
−−
−−
=∫
−
mm
mm
dm
θθ
π
(b) .
22..).........22(2
1)...32)(12(
)(sin
2
0
2
−
−−
=∫ mm
mm
dm π
θθ
π
(or)
16. (a) Explain .)1(
1
0
dxxx pm
−∫ in terms of Beta function and hence
evaluate ( ) .1 2
11
0
2
3
dxxx −∫
(b) Evaluate dxex x2
0
−
∞
∫ in terms of Gamma function.
(a) Prove that β(m,n) = )(
)().(
nm
nm
+Γ
ΓΓ
.
(b) Evaluate ∫
2π
o
cosm
x sinn
x dx, where m of n is even
integers.
(or)
16. (a) Prove that (n+ ½)= )1(.2
)).(12(
12
+Γ
Π+Γ
−
n
n
n

6. (b) Prove that β(n,n) = )(2
)(
2
112
+Γ
ΓΠ
−
n
n
n .
(a) Prove that β(m, n) = m n/
m+n
(b) Evaluate ∫ −
1
0
)1( dxxx pnm
in terms of Gamma function.
(a) Evaluate ( ) dxxx
pnm
−∫ 1
1
0
in terms of Gamma functions and hence find
∫ −
1
0
.
1 n
x
dx
(b) Show the volume of the region of space bounded by the co-
ordinate planes and the surface .
90
1
abc
is
c
z
b
y
a
x =++
(or)
16. (a) Prove that ( )
( )
( )2
12
,
12
+
=
−
n
n
nn
n
π
β

7. (b) Prove that β(n,n) = )(2
)(
2
112
+Γ
ΓΠ
−
n
n
n .
(a) Prove that β(m, n) = m n/
m+n
(b) Evaluate ∫ −
1
0
)1( dxxx pnm
in terms of Gamma function.
(a) Evaluate ( ) dxxx
pnm
−∫ 1
1
0
in terms of Gamma functions and hence find
∫ −
1
0
.
1 n
x
dx
(b) Show the volume of the region of space bounded by the co-
ordinate planes and the surface .
90
1
abc
is
c
z
b
y
a
x =++
(or)
16. (a) Prove that ( )
( )
( )2
12
,
12
+
=
−
n
n
nn
n
π
β