March 2011 - Career Point

Where I´(α) is the derivative of I(α) w.r.t. α and
f ´(x, α) is the derivative of f(x, α) w.r.t. α, kepping x
constant.
Integrals with Infinite Limits :
If a function f(x) is continuous for a ≤ x < ∞, then by
definition
∫ ∞ b
f ( x)
dx = lim
a
b→∞∫
f ( x)
dx ....(i)
a
If there exists a finite limit on the right hand side of
(i), then the improper integrals is said to be
convergent; otherwise it is divergent.
Geometrically, the improper integral (i) for f(x) > 0,
is the area of the figure bounded by the graph of the
function y = f(x), the straight line x = a and the x-axis.
Similarly,
∫ ∞ −
b
∫ ∞ −∞
( dx = lim
a→−∞∫
f x)
properties :
∫ a
0
and
∫
∫ π / 2
0
a
f (x) dx =
∫ −∞
b
a
f ( x)
dx and
f ( x)
dx +
∫ ∞ a
f ( x)
dx
1
x f ( x)
dx = a 2 ∫ a x f ( x)
if f(a – x) = f(x)
a
f ( x)
f ( x)
+ f ( a − x
0 )
0
dx = 2
a
logsin
x dx =
∫ π / 2
logcos
x dx
0
= – 2
π log 2 = 2
π log 2
1
⎛ 1 ⎞
Γ(n + 1) = n Γ (n), Γ(1) = 1, Γ ⎜ ⎟ = π
⎝ 2 ⎠
If m and n are non-negative integers, then
∫ π / 2
0
sin
m
x cos
n
⎛ m + 1⎞
⎛ n + 1⎞
Γ⎜
⎟Γ⎜
⎟
x dx =
⎝ 2 ⎠ ⎝ 2 ⎠
⎛ m + n + 2 ⎞
2Γ⎜
⎟
⎝ 2 ⎠
Reduction Formulae of some Define Integrals :
∫ ∞
0
∫ ∞
0
∫ ∞
0
−ax
e cos bx dx =
−ax
e sin bx dx =
−ax
e x n dx =
If I n =
∫ π / 2
0
sin
n
a
n!
n+ 1
x dx
a
a
2
2
a
+ b
b
+ b
, then
2
2
⎧ n −1
n − 3 n − 5 2
⎪ . . .....
I n = n n − 2 n − 4 3
⎨
n −1
n − 3 n − 5 1 π
⎪ . . ..... .
⎩ n n − 2 n − 4 2 2
If I n =
∫ π / 2
0
cos
n
x dx , then
⎧ n −1
n − 3 n − 5 2
⎪ . . .....
I m = n n − 2 n − 4 3
⎨
n −1
n − 3 n − 5 1 π
⎪ . . ..... .
⎩ n n − 2 n − 4 2 2
( when n is odd)
( when n is even)
( when n is odd)
( when n is even)
Leibnitz's Rule :
If f(x) is continuous and u(x), v(x) are differentiable
functions in the interval [a, b], then
d
dx
∫
v(
x)
u(
x)
f ( t)
dt
d d
= f{v(x)} v(x) – f{u(x) u(x)
dx
dx
Summation of Series by Integration :
n
∑ − 1
lim
n→∞
r=
0
⎛ r ⎞ 1
f ⎜ ⎟ . =
⎝ n ⎠ n ∫ 1 f ( x)
dx
0
Some Important Results :
n
∑ − 1
r=
0
n
∑ − 1
r=
0
2
1
sin( α + r β)
=
cos( α + r β)
=
2
1 1 1 π
–
2 +
2 – .... =
2 3 12
1 1 +
2 +
2 + .... =
2 31
6
2
1
⎧ 1 ⎫ ⎛ 1 ⎞
sin⎨α + ( n −1)
β⎬sin⎜
nβ⎟
⎩ 2 ⎭ ⎝ 2 ⎠
⎛ 1 ⎞
sin⎜
β⎟
⎝ 2 ⎠
⎧ 1 ⎫ ⎛ 1 ⎞
cos⎨α + ( n −1)
β⎬sin⎜
nβ⎟
⎩ 2 ⎭ ⎝ 2 ⎠
⎛ 1 ⎞
sin⎜
β⎟
⎝ 2 ⎠
Area under Curves :
Area bounded by the curve y = f(x), the x-axis and the
ordinates x = a, x = b
=
∫ b y dx =
a ∫ b a
Y
f ( x)
dx
2
π
y = f (x)
y
x = b
O δx X
Area bounded by the curve x = f(y), the y-axis and the
abscissae y = a, y = b
XtraEdge for IIT-JEE 47 MARCH 2011