Higher Engineering Mathematics

REDUCTION FORMULAE 427

H

Problem 6. Use a reduction formula to deter-

mine

∫

x^3 sinxdx.

Using equation (3),
∫
x^3 sinxdx=I 3

=−x^3 cosx+ 3 x^2 sinx−3(2)I 1

and I 1 =−x^1 cosx+ 1 x^0 sinx

=−xcosx+sinx

Hence
∫
x^3 sinxdx=−x^3 cosx+ 3 x^2 sinx

−6[−xcosx+sinx]

=−x^3 cosx+ 3 x^2 sinx

+ 6 xcosx−6 sinx+c

Problem 7. Evaluate

∫π

2

0

3 θ^4 sinθdθ, correct

to 2 decimal places.

From equation (3),

In=[−xncosx+nxn−^1 (sinx)]

π

2

0 −n(n−1)In−^2

=

[(

−

(π

2

)n

cos

π

2

+n

(π

2

)n− 1

sin

π

2

)

−(0)

]

−n(n−1)In− 2

=n

(π

2

)n− 1

−n(n− 1 )In− 2

Hence

∫ π

2

0

3 θ^4 sinθdθ= 3

∫π

2

0

θ^4 sinθdθ

= 3 I 4

= 3

[

4

(π

2

) 3

−4(3)I 2

]

I 2 = 2

(π

2

) 1

−2(1)I 0 and

I 0 =

∫π

2

0

θ^0 sinθdθ=[−cosx]

π

2

0

=[− 0 −(−1)]= 1

Hence

3

∫π

2

0

θ^4 sinθdθ

= 3 I 4

= 3

[

4

(π

2

) 3

−4(3)

{

2

(π

2

) 1

−2(1)I 0

}]

= 3

[

4

(π

2

) 3

−4(3)

{

2

(π

2

) 1

−2(1)(1)

}]

= 3

[

4

(π

2

) 3

− 24

(π

2

) 1

+ 24

]

=3(15. 503 − 37. 699 +24)

=3(1.8039)= 5. 41

Now try the following exercise.

Exercise 171 Further problems on reduc-

tion formulae for integrals of the form∫

xncosxdxand

∫

xnsinxdx

1. Use a reduction formula to determine∫
   x^5 cosxdx.
   ⎡

⎣

x^5 sinx+ 5 x^4 cosx− 20 x^3 sinx

− 60 x^2 cosx+ 120 xsinx

+120 cosx+c

⎤

⎦

2. Evaluate

∫π

0 x

(^5) cosxdx, correct to 2 decimal
places. [−134.87]

3. Use a reduction formula to determine∫
   x^5 sinxdx.
   ⎡

⎣

−x^5 cosx+ 5 x^4 sinx+ 20 x^3 cosx

− 60 x^2 sinx− 120 xcosx

+120 sinx+c

⎤

⎦

4. Evaluate

∫π

0 x

(^5) sinxdx, correct to 2 decimal
places. [62.89]
44.4 Using reduction formulae for
integrals of the form
∫
sinnxdx
and
∫
cosnxdx
(a)
∫
sinnxdx
LetIn=
∫
sinnxdx≡
∫
sinn−^1 xsinxdxfrom laws
of indices.
Using integration by parts, letu=sinn−^1 x, from
which,