Higher Engineering Mathematics, Sixth Edition

Essential formulae 667

∫π

0

xncosxdx=In=−nπn−^1 −n(n− 1 )In− 2

∫
xnsinxdx=In=−xncosx+nxn−^1 sinx

−n(n− 1 )In− 2
∫
sinnxdx=In=−

1

n

sinn−^1 xcosx+

n− 1

n

In− 2

∫
cosnxdx=In=

1

n

cosn−^1 sinx+

n− 1

n

In− 2

∫π/ 2

0

sinnxdx=

∫π/ 2

0

cosnxdx=In=

n− 1

n

In− 2

∫

tannxdx=In=

tann−^1 x

n− 1

−In− 2

∫

(lnx)ndx=In=x(lnx)n−nIn− 1

With reference to Fig. FA4.

y

y 5 f(x)

0 x 5 ax 5 b x

A

Figure FA4

Area under a curve:

areaA=

∫b

a

ydx

Mean value:

mean value=

1

b−a

∫b

a

ydx

R.m.s. value:

r.m.s. value=

√{

1

b−a

∫b

a

y^2 dx

}

Volume of solid of revolution:

volume=

∫b

a

πy^2 dxabout thex-axis

Centroids:

With reference to Fig. FA5:

x ̄=

∫b

a

xydx

∫b

a

ydx

and y ̄=

1

2

∫b

a

y^2 dx

∫b

a

ydx

y

C

Area A

x

y

y 5 f(x)

0 x^5 ax^5 b x

Figure FA5

Theorem of Pappus:

WithreferencetoFig.FA5,whenthecurveisrotatedone

revolutionabout thex-axis between thelimitsx=aand

x=b,thevolumeVgenerated is given by:V= 2 πAy ̄.

Parallel axis theorem:

IfCis the centroid of areaAin Fig. FA6 then

Ak^2 BB=AkGG^2 +Ad^2 ork^2 BB=kGG^2 +d^2

G B

G B

d

C

Area A

Figure FA6