Higher Engineering Mathematics

Ess-For-H8152.tex 19/7/2006 18: 2 Page 712

712 ESSENTIAL FORMULAE

Integral Calculus

Standard integrals

y

∫

ydx

axn a

xn+^1

n+ 1

+c

(except wheren=−1)

cosax

1

a

sinax+c

sinax −

1

a

cosax+c

sec^2 ax

1

a

tanax+c

cosec^2 ax −

1

a

cotax+c

cosecaxcotax −

1

a

cosecax+c

secaxtanax

1

a

secax+c

eax

1

a

eax+c

1

x

lnx+c

tanax

1

a

ln ( secax)+c

cos^2 x

1

2

(

x+

sin 2x

2

)

+c

sin^2 x

1

2

(

x−

sin 2x

2

)

+c

tan^2 x tanx−x+c

cot^2 x −cotx−x+c

1

√

(a^2 −x^2 )

sin−^1

x

a

+c

√

(a^2 −x^2 )

a^2

2

sin−^1

x

a

+

x

2

√

(a^2 −x^2 )+c

1

(a^2 +x^2 )

1

a

tan−^1

x

a

+c

y

∫

ydx

1

√

(x^2 +a^2 )

sinh−^1

x

a

+cor

ln

[

x+

√

(x^2 +a^2 )

a

]

+c

√

(x^2 +a^2 )

a^2

2

sinh−^1

x

a

+

x

2

√

(x^2 +a^2 )+c

1

√

(x^2 −a^2 )

cosh−^1

x

a

+cor

ln

[

x+

√

(x^2 −a^2 )

a

]

+c

√

(x^2 −a^2 )

x

2

√

(x^2 −a^2 )−

a^2

2

cosh−^1

x

a

+c

t=tan

θ

2

substitution

To determine

∫ 1

acosθ+bsinθ+c

dθlet

sinθ=

2 t

(1+t^2 )

cosθ=

1 −t^2

1 +t^2

and

dθ=

2 dt

(1+t^2 )

Integration by parts

Ifuandvare both functions ofxthen:

∫

u

dv

dx

dx=uv−

∫

v

du

dx

dx

Reduction formulae

∫

xnexdx=In=xnex−nIn− 1

∫

xncosxdx=In=xnsinx+nxn−^1 cosx

−n(n−1)In− 2