Higher Engineering Mathematics

Ess-For-H8152.tex 19/7/2006 18: 2 Page 713

ESSENTIAL FORMULAE 713

∫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.

0 x  ax  b x

y

y  f(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

Area A

y  f(x)

C

y

x

0 x  ax  bx

y

Figure FA5