Higher Engineering Mathematics, Sixth Edition

Reduction formulae 429

Problem 6.∫ Use a reduction formula to determine

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−6sinx+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 170 Further problems on

reduction formulae for integrals of the form∫

xncosxdxand

∫

xnsinxdx

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

⎢

⎣

x^5 sinx+ 5 x^4 cosx− 20 x^3 sinx

− 60 x^2 cosx+ 120 xsinx

+120cosx+c

⎤

⎥

⎦

2. Evaluate

∫π

0 x

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

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

⎢

⎣

−x^5 cosx+ 5 x^4 sinx+ 20 x^3 cosx

− 60 x^2 sinx− 120 xcosx

+120sinx+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.

Usingintegrationbyparts,letu=sinn−^1 x,fromwhich,