604
Appendix: Problem Index
1.2.3 Gamma and Beta Function
(a) Γ(Z + 1) = Z Γ(Z ) (b) Γ(n + 1) = n! for positive integer
1.23
β(m, n) = Γ(m)Γ(n)/Γ(m + n)
1 . 2 4
To evaluate a definite integral using β functions
1.25
(a) Γ(n)Γ(1 − n) = π/ sin(nπ); 0 < n < 1
(b) |Γ(in)|
2
= π/n sinh(nπ)
1.26
1.2.4 Matrix Algebra
Characteristic roots of Hermitian matrix are real
1.27
Characteristic equation and eigen values of the given matrix
1.28
Effect of a set of matrices on position vector
1.29
Eigen values of the given matrix
1.30
Diagnalization of a matrix and its trace
1.31
Eigen values, eigen vector, modal matrix C and its inverse C
−1 ,
product C
−1 AC
1.32
1.2.5 Maxima and Minima
Solution of a cubic equation by Newton’s method
1.33
(a) Turning points of f(x) (b) Whether f(x) is odd or even or neither 1.34
1.2.6 Series
Interval of convergence of series
1.35
Expansion of log x by Taylor’s series
1.36
Expansion of cos x into infinite power series
1.37
Expansion of sin(a + x) by Taylor’s series
1.38
Sum of series 1 + 2x + 3x
2
+ 4x
3
+ . . . , |x| < 1
1 . 3 9
1.2.7 Integration
(a)
sin
3 x cos
6 x dx (b)
sin
4 x cos
2 x dx
1.40
(2x
2
− 3x − 2)
−1 dx
1.41
(a) Sketch of curve r
2
= a
2 sin 2θ (b) area within the curve between
θ = 0 and θ = π/2
1.42
(x
3
+ x
2
+ 2)(x
2
+ 2)
−2 dx
1.43
∞
0 4a
3 (x
2
+ 4a
2 )
−1 dx
1.44
(a)
tan
6 x sec
4 x dx (b)
tan
5 x sec
3 x dx
1.45
4
2 (2x + 4)(x
2
− 4x + 8)
−1 dx = ln 2 + π
1.46
Area included between curve y
2
= x
3 and line x = 4
1 . 4 7
Surface of revolution of hypocycloid about x-axis
1.48
a
0
√
a 2 −x 2
0
(x + y)dydx
1.49
Area enclosed between curves y = 1/x and y = −1/x and lines
x = 1 and x = 2
1.50
(x
2
− 18x + 34)
−1 dx
1.51
1
0 x
2 tan
−1 x dx
1.52
(a) Area bounded by curves y = x
2
+ 2 and y = x − 1 and lines
x = −1 and x = 2
(b) Volume of solid of revolution obtained by rotating area enclosed
by lines x = 0, y = 0, x = 2 and 2x + y = 5 through 2π rad about
y-axis
1.53
Appendix: Problem Index
1.2.3 Gamma and Beta Function
(a) Γ(Z + 1) = Z Γ(Z ) (b) Γ(n + 1) = n! for positive integer
1.23
β(m, n) = Γ(m)Γ(n)/Γ(m + n)
1 . 2 4
To evaluate a definite integral using β functions
1.25
(a) Γ(n)Γ(1 − n) = π/ sin(nπ); 0 < n < 1
(b) |Γ(in)|
2
= π/n sinh(nπ)
1.26
1.2.4 Matrix Algebra
Characteristic roots of Hermitian matrix are real
1.27
Characteristic equation and eigen values of the given matrix
1.28
Effect of a set of matrices on position vector
1.29
Eigen values of the given matrix
1.30
Diagnalization of a matrix and its trace
1.31
Eigen values, eigen vector, modal matrix C and its inverse C
−1 ,
product C
−1 AC
1.32
1.2.5 Maxima and Minima
Solution of a cubic equation by Newton’s method
1.33
(a) Turning points of f(x) (b) Whether f(x) is odd or even or neither 1.34
1.2.6 Series
Interval of convergence of series
1.35
Expansion of log x by Taylor’s series
1.36
Expansion of cos x into infinite power series
1.37
Expansion of sin(a + x) by Taylor’s series
1.38
Sum of series 1 + 2x + 3x
2
+ 4x
3
+ . . . , |x| < 1
1 . 3 9
1.2.7 Integration
(a)
sin
3 x cos
6 x dx (b)
sin
4 x cos
2 x dx
1.40
(2x
2
− 3x − 2)
−1 dx
1.41
(a) Sketch of curve r
2
= a
2 sin 2θ (b) area within the curve between
θ = 0 and θ = π/2
1.42
(x
3
+ x
2
+ 2)(x
2
+ 2)
−2 dx
1.43
∞
0 4a
3 (x
2
+ 4a
2 )
−1 dx
1.44
(a)
tan
6 x sec
4 x dx (b)
tan
5 x sec
3 x dx
1.45
4
2 (2x + 4)(x
2
− 4x + 8)
−1 dx = ln 2 + π
1.46
Area included between curve y
2
= x
3 and line x = 4
1 . 4 7
Surface of revolution of hypocycloid about x-axis
1.48
a
0
√
a 2 −x 2
0
(x + y)dydx
1.49
Area enclosed between curves y = 1/x and y = −1/x and lines
x = 1 and x = 2
1.50
(x
2
− 18x + 34)
−1 dx
1.51
1
0 x
2 tan
−1 x dx
1.52
(a) Area bounded by curves y = x
2
+ 2 and y = x − 1 and lines
x = −1 and x = 2
(b) Volume of solid of revolution obtained by rotating area enclosed
by lines x = 0, y = 0, x = 2 and 2x + y = 5 through 2π rad about
y-axis
1.53