reduction formulas - MadAsMaths · Created by T. Madas Created by T. Madas Question 25 (****+) ( )...
Transcript of reduction formulas - MadAsMaths · Created by T. Madas Created by T. Madas Question 25 (****+) ( )...
Created by T. Madas
Created by T. Madas
REDUCTION
FORMULAS
Created by T. Madas
Created by T. Madas
Question 1 (**)
21
0
12e
xnnI x dx
−= , n ∈� .
Show clearly that…
a) … ( ) 2
121 en nI n I
−
−= − − , 2n ≥ .
b) … 21
5
0
1 12 2e 8 13e
xx dx
− −= − .
proof
Created by T. Madas
Created by T. Madas
Question 2 (**)
0
2sinn
nI x dx
π
= , n ∈� .
Show clearly that…
a) … 2
1n n
nI I
n−
−= , 2n ≥ .
b) … ( )6 2
0
2 45sin 1 cos
256x x dx
π
π+ = .
proof
Created by T. Madas
Created by T. Madas
Question 3 (**)
( )
12
0
1 2 en x
nI x dx= − , n ∈� .
Show clearly that…
a) …. 12 1n nI nI −= − , 1n ≥ .
b) … 4 384 e 633I = − .
proof
Created by T. Madas
Created by T. Madas
Question 4 (**)
0
2cosn
nI x dx
π
= , n ∈� .
Show clearly that…
a) … 2
1n n
nI I
n−
−= , 2n ≥ .
b) … 5 2
0
2 8cos sin
105x x dx
π
= .
proof
Created by T. Madas
Created by T. Madas
Question 5 (**+)
( )e
1
lnn
nI x x dx= , n ∈� .
Show clearly that…
a) … 212 en nI nI −= − .
b) … ( )24
1e 3
4I = − .
proof
Created by T. Madas
Created by T. Madas
Question 6 (**+)
( )e
2
1
lnn
nI x x dx= , n ∈� .
a) Show clearly that
( )31
1e
3n nI nI −= − .
The part of the curve with equation ( )2
3 lny x x= , for 1 ex≤ ≤ , is rotated by 2π
radians about the x axis.
b) Show that the volume of the solid generated is given by
( )311e 89
π− .
proof
Created by T. Madas
Created by T. Madas
Question 7 (**+)
secnnI x dx= , n ∈� .
Show clearly that
( )( )2
2
1 2tan sec
1 1
nn n
nI x x I
n n
−
−
−= +
− −, 2n ≥ .
proof
Created by T. Madas
Created by T. Madas
Question 8 (***)
arsinh1
0
sinhnnI x dx= , n ∈� .
Show clearly that…
a) … ( ) 22 1n nnI n I −= − − , 2n ≥ .
b) … ( )5
17 2 8
15I = − .
proof
Created by T. Madas
Created by T. Madas
Question 9 (***)
The integral nI is defined for 0n ≥ as
4
0
tannnI x dx
π
= , n ∈� .
Show clearly that…
a) … 2
1
1n nI I
n−= −
−, 1n ≥ .
b) … ( )4
13 8
12I π= − .
proof
Created by T. Madas
Created by T. Madas
Question 10 (***)
0
tanh
an
nI x dx= , n ∈� .
Given that 1artanh2
a = , show clearly that
( )1
2
0.5
1
n
n nI In
−
−= −−
, 2n ≥ .
proof
Created by T. Madas
Created by T. Madas
Question 11 (***+)
By forming and using a suitable reduction formula, or otherwise, show that
21
5
0
2e 5e
2e
xx dx
− −= .
proof
Created by T. Madas
Created by T. Madas
Question 12 (***+)
The integral nI is defined for 0n ≥ as
0
2cosn
nI x x dx
π
= .
a) Show clearly that …
… ( ) 212
n
n nI n n Iπ
−
= − −
, 2n ≥ .
b) Hence find, in terms of π , exact expressions for …
i. … 4
4 2
0
2cos 3 24
16x x dx
ππ
π= − + .
ii. … 4
5 2
0
2 5sin 15 120
16x x dx
ππ
π= − +
44 2
0
2
cos 3 2416
x x dx
π
ππ= − + ,
45 2
0
2 5sin 15 120
16x x dx
π
ππ= − +
Created by T. Madas
Created by T. Madas
Question 13 (***+)
The integral nI is defined for 0n ≥ as
( )32
1
0
1nnI x x dx= − , n ∈� .
Show that
1
2
2 5n n
nI I
n−
=
+ , 1n ≥ ,
and use it to find as an exact fraction the value of 3I .
332
1155I =
Created by T. Madas
Created by T. Madas
Question 14 (***+)
ln 2
0
tanhnnI x dx= , n ∈� .
Show clearly that …
a) …
1
2
1 3
1 5
n
n nI In
−
−
= −
− , 2n ≥ .
b) …
2
1
1 3 5ln
2 5 4
r
r
r
∞
=
=
.
proof
Created by T. Madas
Created by T. Madas
Question 15 (****)
( )1
2
0
1n
nI x dx= − , n ∈� .
Show clearly that
1
2
2 1n n
nI I
n−=
+, 1n ≥ .
proof
Created by T. Madas
Created by T. Madas
Question 16 (****)
1
20
1
n
n
xI dx
x=
+ , n ∈� .
Show clearly that…
a) … ( ) 22 1n nnI n I −= − − , 2n ≥ .
b) … ( )1
3
20
12 2
31
xdx
x= −
+ .
proof
Created by T. Madas
Created by T. Madas
Question 17 (****)
( )3
2
0
3n
nI x dx= − , n ∈� .
Show clearly that…
a) … ( ) 12 1 6n nn I nI −+ = , 1n ≥ .
b) … ( )3
42
0
11523 3
35x dx− = .
proof
Created by T. Madas
Created by T. Madas
Question 18 (****)
12
0
1nnI x x dx= − , n ∈� .
Show clearly that…
a) … ( ) ( ) 22 1n nn I n I −+ = − , 2n ≥ .
b) …
12
0
161
315
nx x dx− = .
proof
Created by T. Madas
Created by T. Madas
Question 19 (****)
( )21n
nI x dx−
= + , n ∈� .
Show clearly that
( )2
1
1 2 1
2 2
n
n n
x x nI I
n n
−
+
+ −= + , 1n ≥ .
proof
Created by T. Madas
Created by T. Madas
Question 20 (****)
The integral nI is defined for 0n ≥ as
0
sinnnI d
π
θ θ θ= .
Show clearly that …
a) … ( ) 21nn nI n n Iπ −= − − , 2n ≥ .
b) … ( )4 4 2
0
2 1sin 2 12 48
32x x dx
π
π π= − + .
proof
Created by T. Madas
Created by T. Madas
Question 21 (****)
21
n
n
xI dx
x=
+ , n ∈� .
a) Find an expression for
( )121 2 1nd
x xdx
− +
.
b) Use part (a) to show that
( ) 1 221 1n
n nnI n I x x−
−+ − = + , 2n ≥ .
( ) ( ) ( ) ( )1 1 12 2 21 2 2 2 21 1 1 1n n nd
x x n x x x xdx
−− −
+ = − + + +
Created by T. Madas
Created by T. Madas
Question 22 (****+)
2e sinx nnI x dx≡ , n ∈� , 2n ≥ .
Use integration by parts twice to show
( ) ( ) ( )2 2 124 1 2sin cos e sinx n
n nn I n n I x n x x−
−+ = − + − .
SPX-G , proof
Created by T. Madas
Created by T. Madas
Question 23 (****+)
Find a suitable reduction formula and use it to find
( )1
10
0
lnx x dx .
You may assume that the integral converges.
Give the answer as the product of powers of prime factors.
( )1
10 3 4 2
0
ln 2 3 5 7x x dx−
= × × ×
Created by T. Madas
Created by T. Madas
Question 24 (****+)
sin
sinn
nxI dx
x= , n ∈� .
a) Show by considering 2n nI I+ − that
( )2
2sin 1
1n nI I n x C
n+ = + + +
+, 0n ≥ .
b) Show further that
( )3
4
sin 6 112 3 17 2
sin 15
xdx
x
π
π= − .
proof
Created by T. Madas
Created by T. Madas
Question 25 (****+)
( )
0
sin
sinn
nI d
πθ
θθ
= .
The integral above is defined for positive integer values n .
a) Use trigonometric identities to show that
( ) ( )( )
sin sin 22cos 1
sin
n nn
θ θθ
θ
− − = − .
b) Hence show that
2n nI I −= , 2n ≥ .
c) Evaluate nI in both cases, where n is either odd or even positive integer.
0 if is even
if is oddn
nI
nπ
=
Created by T. Madas
Created by T. Madas
Question 26 (****+)
3
0
3e tanx n
nI x dx
π
= , n ∈� .
a) Show clearly that…
i. … ( )1 1e 3 3n
n n nnI I nIπ
+ −= − − , 1n ≥ .
ii. … 0 4 3 13I I I I= + − .
b) Hence find the exact value of
( )3 3 2
0
3e tan tan sec 4x
x x x dx
π
+ − .
proof
Created by T. Madas
Created by T. Madas
Question 27 (****+)
2
,0
sin cosm nm nI d
π
θ θ θ= , m ∈� , n ∈� .
a) Show clearly that
, 2,
1m n m n
mI I
m n−
−=
+.
b) Hence find an exact value for
22
0
sin sin 2 d
π
θ θ θ .
128
315
Created by T. Madas
Created by T. Madas
Question 28 (*****)
It is given that
12
0
an
nI x a x dx+
= − , n ∈� , 0n ≥
where a is a positive constant.
a) Use integration by parts to show
02 2
4 1
n
n
na II
n n
+ =
+ , 1n ≥ .
b) Determine the value of
210 2
0
4x x dx− .
SPX-E , 42π
Created by T. Madas
Created by T. Madas
Question 29 (*****)
It is given that
( )1
,0
1n m
n mI x x dx= − ,
where ,n m∈� , with , 0n m ≥ .
a) Show that …
i. … , 1, 1, 1n m n m n mI I I− − +− = − .
ii. … , 1, 1n m n m
nI I
m− +=
b) Hence derive an expression of ,n mI and use it to find
( )12
13
0
7 1x x dx− .
SPX-C , ( )12
13
0
327 1
45x x dx− =
Created by T. Madas
Created by T. Madas
Question 30 (*****)
( ) ( ) ( ),
b
m n
a
I m n b x x a dx= − − , m∈� , n∈� .
Show that
( )( )
( )1! !
,1 !
m nm nI m n b a
m n
+ += −
+ +,
where a and b are real constants such that b a>
SPX-B , proof
Created by T. Madas
Created by T. Madas
Question 31 (*****)
2 20
an
n
xI dx
a x=
− , n ∈� , 0a > .
Show clearly that…
a) … ( )2
2
1n n
a nI I
n−
−= , 2n ≥ .
b) …
43 2
22
3 18 36 183 16
4
x x xdx
x xπ
− + −= −
− .
FP4-S , proof