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Reduction Formula

1. The document presents several integrals and asks to prove reduction formulas relating integrals of different orders. It asks to use the reduction formulas to evaluate specific integrals and deduce further formulas and results. 2. Reduction formulas presented relate integrals of the form In to In-1 and In-2, allowing integrals of higher order to be evaluated recursively in terms of those of lower order. 3. Several results are also deduced from the reduction formulas, including evaluations of specific integrals and generalized formulas for families of integrals.

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Kuldeep Lamba
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© © All Rights Reserved
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834 views2 pages

Reduction Formula

1. The document presents several integrals and asks to prove reduction formulas relating integrals of different orders. It asks to use the reduction formulas to evaluate specific integrals and deduce further formulas and results. 2. Reduction formulas presented relate integrals of the form In to In-1 and In-2, allowing integrals of higher order to be evaluated recursively in terms of those of lower order. 3. Several results are also deduced from the reduction formulas, including evaluations of specific integrals and generalized formulas for families of integrals.

Uploaded by

Kuldeep Lamba
Copyright
© © All Rights Reserved
We take content rights seriously. If you suspect this is your content, claim it here.
Available Formats
Download as PDF, TXT or read online on Scribd
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Reduction Formula

1.

x 2n

In =

(a)

Prove that

(i)

(b)

By considering

Jn + Jn-1

1 x2

dx

J0 =

2.

If

un =

If

In =

/ 2

Evaluate
3.

If

If

1 x2

(ii)

and

xn

a +x

, show that the reduction formula for

In

allows

Jn

to be evaluated

n.

74 2

16

.
un =

1
n

n 1
n

u n 2 .

u5 .

dx

x5
5 + x2

( n = 0, 1, 2, 3, )

dx

2nIn = (2n 1) In-1 .

x cos n xdx , where n > 1 , show that

u4

In =

, show that

x n 1
n

a2 + x2

n 1
n

a 2 I n 2 , where

n2.

dx .

u m = x m (a 2 x 2 ) dx , show that (m + 2) um = xm 1 (a2 x2 )3/2 + a2 (m 1) um 2 .

1/ 2

/ 2

Evaluate
5.

Evaluate
4.

J3 =

Prove that

x 2n

I0

for any particular value of


(c)

(1 + x )

Jn =

and

sin 2 m cos 2 d , where m is a positive integer.

u n = x n (2ax x 2 ) dx , where n is a positive integer, prove that:

1/ 2

(n + 2) un (2n + 1) aun 1 + x n 1 (2ax x2)3/2 = 0 .

Show that
6.

7.

If

If

In =

x 2 (2ax x 2 ) dx =

2a

1/ 2

xn

3a + x
2

I p ,q =

xp

(1 + x )

2 q

5a 4

dx , show that

(1 + x )

2 3

In =

2a
0

Let

In (z) =

3( n 1)
n

a 2 I n 2 .

x p1

(1 + x 2 )q1

Evaluate

I7 .

+ ( p 1)I p2 ,q 1 .

dx .

x n 2ax x 2 dx , prove that 3aIn 3In+1 = n (In+1 2a In) .

Hence, or otherwise, show that

9.

2a n

2(q 1)I p ,q =
x6

If

In =

dx , show that

Hence, or otherwise, evaluate

8.

1
0

Prove that for all

In =

(1 y) n (e yz 1)dy ,
n 1,

I n 1 ( z ) =

z
n

( 2 n + 1)( 2 n 1)...7 5 n+2


a
, where
( n + 2)( n + 1)...5 4 2
for all

I n (z) +

is a positive integer.

n0.

z
n ( n + 1)

. Deduce that

ez =

zr

r!
r =0

zn
( n 1)!

I n 1 ( z ) .
1

10.

11.

(a)

(b)

Prove that

If

In =

(a)

(ii)

(iii)

(c)

13.

(a)
(b)

14.

If

x m (ln x ) n dx =

(1 x ) dx
1

2 n

and use it to evaluate

2 8

( 1)n n!
.
( m + 1)n +1

x n (a 2 x 2 )1 / 2 dx ,

n > 1,

n 1 2
In =
a I n 2 .
n +2

prove that

I4 .

Prove the Wallis formulae:


(i)

(b)

(1 x ) dx

Hence find

12.

In =

Find a reduction formula for the integral

/2

/2

/2

sin 2 n x dx =

2 2 n n! n! 2

sin 2 n+1 x dx =

(2 n )!
2 2 n n! n!

(2 n + 1)!

sin 2 n 1 x dx = 1 +

2n
0<x<

Prove that

Deduce that

/ 2

, then

I ( p, q ) =

/2

/2

/2
0

sin 2 n x dx = 1

sin 2 n+1 xdx

Obtain a reduction formula for


Prove that

sin 2 n x dx = 1 + n

2n

1 n

2n + 1 n 2

/4
0

sin x cos x

sin x + cos x

/2

sin 2 n +1 xdx ,

where

and deduce that

( 2 n )!
2n

2 n!n!

= 1

0 < n < 1 .

1 n

2n + 1 n

2 n +1

dx

and hence deduce its value.

(ln cos x ) cos xdx = ln 2 1 .

( x a ) p (b x ) q dx

(b > a),

prove that when

n1,

(i)

I(n, n 1) = I(n 1, n)

(ii)

2(2n + 1) I(n, n) = 2n (b a) I(n , n 1)


= n(b a)2 I(n 1, n 1).
Hence or otherwise, show that

15.

Find a reduction formula for

(5 + x )
2

Prove that

2 3/ 2

(a

dx =

I( n , n ) =

+ x2 )

n/2

( 2 n + 1)!

dx , where n is an odd positive integer.

396 + 75 ln 5
16

( b a )2 n +1 n!n!

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