432 Higher Engineering Mathematics
Hence
∫
cos^4 xdx
=
1
4
cos^3 xsinx+
3
4
(
1
2
cosxsinx+
1
2
x
)
=
1
4
cos^3 xsinx+
3
8
cosxsinx+
3
8
x+c
Problem 12. Determine a reduction formula
for
∫ π 2
0
cosnxdxand hence evaluate
∫ π 2
0
cos^5 xdx
From equation (5),
∫
cosnxdx=
1
n
cosn−^1 xsinx+
n− 1
n
In− 2
and hence
∫ π
2
0
cosnxdx=
[
1
n
cosn−^1 xsinx
]π 2
0
+
n− 1
n
In− 2
=[0−0]+
n− 1
n
In− 2
i.e.
∫ π
2
0
cosnxdx=In=
n− 1
n
In− 2 (6)
(Note that this is the same reduction formula as for
∫ π
2
0
sinnxdx(in Problem 10) and the result is usually
known asWallis’s formula).
Thus, from equation (6),
∫ π
2
0
cos^5 xdx=
4
5
I 3 , I 3 =
2
3
I 1
and I 1 =
∫ π
2
0
cos^1 xdx
=[sinx]
π 2
0 =(^1 −^0 )=^1
Hence
∫ π
2
0
cos^5 xdx=
4
5
I 3 =
4
5
[
2
3
I 1
]
=
4
5
[
2
3
( 1 )
]
=
8
15
Now try the following exercise
Exercise 171 Further problems on
reduction formulae for integrals of the form∫
sinnxdxand
∫
cosnxdx
- Use a reduction formula to determine∫
sin^7 xdx.
⎡
⎢
⎣
−
1
7
sin^6 xcosx−
6
35
sin^4 xcosx
−
8
35
sin^2 xcosx−
16
35
cosx+c
⎤
⎥
⎦
- Evaluate
∫π
0 3sin
(^3) xdx using a reduction
formula. [4]
- Evaluate
∫ π
2
0
sin^5 xdx using a reduction
formula.
[
8
15
]
- Determine, using a reduction formula,∫
cos^6 xdx.
⎡
⎢
⎣
1
6
cos^5 xsinx+
5
24
cos^3 xsinx
+
5
16
cosxsinx+
5
16
x+c
⎤
⎥
⎦
- Evaluate
∫ π
2
0
cos^7 xdx.
[
16
35
]
44.5 Further reduction formulae
The following worked problems demonstrate further
examples where integrals can be determined using
reduction formulae.
Problem 13.∫ Determine a reduction formula for
tannxdxand hence find
∫
tan^7 xdx.
LetIn=
∫
tannxdx≡
∫
tann−^2 xtan^2 xdx
by the laws of indices
=
∫
tann−^2 x(sec^2 x− 1 )dx
since 1+tan^2 x=sec^2 x
=
∫
tann−^2 xsec^2 xdx−
∫
tann−^2 xdx