INTEGRATION

FORMULAS

&

EXAMPLES

1)

Ior:u*c

EXAMPLE'

I*:x*c

z)

I

o t(u)ou: kJ

qupu

ExAMpLE,

J,n*'dx:

znJ^ *'&*

3)

J

ft(")

+

g(u)ldu:

.l

r1u;a.,

+

J

g(u)du

(integrate

a sum over each

term)

ExAMPLE'

J

(*'+2x+

3)dx:

f

*'dr.*

J

,*a**

J

,a*

:

"f

*'dx+2J

*a**:J

a,

4)

(POWERRULE):

Jrr'dr:

u'*'

*c

(Mustmakesureduispresent)

(tt*-t)

n+l

EXAMPLE,

J

*'dx:

xo

*c,

since

u::g du:dx

&

is

present.

+

EXAMPLE'

J

1S*'+

7)'xdx,

since

u:5x'+7,

du:l0xdx

(need

to multiply inside

by l0

& compensate

by dividing

on the

outside

by 10).

(1/10)

I

1r*'+7;'10xdx:

(5x'+7)'

*

c

:

(sx'

+7)'

190

+

c

(r0xe)

5)

J

.'d,r:

e'+ c, EXAMPLET

J

*"dx:

(1/5)J

""5dx:

(1/5)e"+

c

6)

J

or,r:hlrrl+.,

EX:

J

lrin:*yd:/cos3x:G1/3)J

1-ri**y3d:r/cos3x:(-rl3)lnlcos3xl+c

7)

J

.os(r)du:

sin(u)

+c

(8)

J

sin(u)du

:

-cos(u)

+c

(9)

J

sec'udu:

tan(u)

+c

EXAMPLE

of

7):

J

11cos1lnx)l

/x)

dx:

J

;.or1kx)l

[1/x]

dx: sin(hx)

+

c