Integration Formulas



Integration Formulas

1. Common Integrals

Indefinite Integral

Method of substitution

f (g(x))g(x)dx = f (u)du

Integration by parts

f (x)g(x)dx = f (x)g(x) - g(x) f (x)dx

Integrals of Rational and Irrational Functions

xndx = xn+1 + C n +1

1dx = ln x + C x

c dx = cx + C

xdx = x2 + C

2

x2dx = x3 + C

3

1 x2

dx

=

-

1 x

+

C

xdx = 2x x + C 3

1

1 + x2

dx

=

arctan

x

+

C

1 dx = arcsin x + C 1- x2

Integrals of Trigonometric Functions

sin x dx = - cos x + C

cos x dx = sin x + C

tan x dx = ln sec x + C

sec x dx = ln tan x + sec x + C

sin2 x dx = 1 ( x - sin x cos x) + C 2

cos2 x dx = 1 ( x + sin x cos x) + C 2

tan2 x dx = tan x - x + C

sec2 x dx = tan x + C

Integrals of Exponential and Logarithmic Functions

ln x dx = x ln x - x + C

xn ln x dx = xn+1 ln x - xn+1 + C

n +1

(n + 1)2

ex dx = ex + C

bx dx = bx + C ln b

sinh x dx = cosh x + C

cosh x dx = sinh x + C



2. Integrals of Rational Functions

Integrals involving ax + b

(

ax

+

b)n

dx

=

(ax + a(n

b)n+1 + 1)

( for n -1)

1 dx = 1 ln ax + b ax + b a

x

( ax

+

b)n

dx

=

a (n + 1) x - b a2 (n +1)(n + 2)

(ax

+

b)n+1

( for n -1, n -2)

x ax +

b

dx

=

x a

-

b a2

ln

ax

+

b

(ax

x +

b)2

dx

=

a2

b

( ax

+

b)

+

1 a2

ln

ax

+

b

(ax

x +

b)n

dx

=

a2

(n

a (1 - n) x - b -1)(n - 2)(ax +

b)n-1

( for n -1, n -2)

x2 ax + b

dx

=

1 a3

(

ax

+

b

)2

2

- 2b(ax + b) + b2

ln

ax + b

x2

(ax + b)2

dx

=

1 a3

ax

+

b

-

2b ln

ax

+

b

-

b2 ax +

b

x2

(ax + b)3

dx

=

1 a3

ln

ax

+

b

+

2b ax +

b

-

b2

2(ax +

b)2

x2

(ax + b)n

dx

=

1 a3

-

(

ax + b)3-n

n-3

+

2b (a + b)2-n

n-2

-

b2

(ax + b)1-n

n -1

( for n 1, 2,3)

1

x (ax + b)

dx

=

-

1 b

ln

ax + x

b

x2

1 dx

(ax + b)

=

-1 bx

+

a b2

ln

ax + b x

1

x2 (ax + b)2

dx

=

-a

b2

(

1 a+

xb)

+

1 ab2 x

-

2 b3

ln

ax + b x

Integrals involving ax2 + bx + c

1 dx = 1 arctg x

x2 + a2

a

a

x2

1 -

a2

dx

=

1

2a

1

2a

ln ln

a a x x

- + - +

x x a a

for x < a for x > a



2 arctan

4ac - b2

2ax + b 4ac - b2

ax2

1 + bx

+

c

dx

=

2 ln 2ax + b - b2 - 4ac 2ax + b +

b2 - 4ac b2 - 4ac

-

2 2ax +

b

for 4ac - b2 > 0 for 4ac - b2 < 0 for 4ac - b2 = 0

ax2

x + bx

+

c

dx

=

1 2a

ln

ax2

+ bx

+

c

-

b 2a

ax2

dx + bx

+

c

m

ln

ax2

+ bx

+c

+

2an - bm

arctan

2ax + b

for 4ac - b2 > 0

2a

a 4ac - b2

4ac - b2

mx + n ax2 + bx +

c

dx

=

m 2a

ln

ax2

+ bx

+

c

+

2an - bm a b2 - 4ac

arctanh

2ax + b b2 - 4ac

for 4ac - b2 < 0

m

2a

ln

ax2

+ bx

+c

-

2an - bm

a (2ax + b)

for 4ac - b2 = 0

( ) ( )( ) ( ) ( ) 1 ax2 + bx + c

n

dx =

(n -1)

2ax + b 4ac - b2 ax2

+ bx + c

n-1

(2n - 3) 2a

+

(n -1) 4ac - b2

1 ax2 + bx + c n-1 dx

( ) 1

dx = 1 ln

x2

-b

1

dx

x ax2 + bx + c

2c ax2 + bx + c 2c ax2 + bx + c

3. Integrals of Exponential Functions

xecxdx = ecx (cx - 1) c2

x2ecxdx

=

ecx

x2 c

-

2x c2

+

2 c3

xnecx dx = 1 xnecx - n xn-1ecx dx

c

c

ecx dx = ln x + (cx)i

x

i=1 i i!

ecx

ln

xdx

=

1 ecx c

ln

x

+

Ei

(cx)

ecx sin bxdx = ecx (c sin bx - b cos bx) c2 + b2

ecx cos bxdx = ecx (c cos bx + bsin bx) c2 + b2

ecx sinn xdx = ecx sinn-1 x (c sin x - n cos bx) + n (n -1) ecx sinn-2 dx

c2 + n2

c2 + n2



4. Integrals of Logarithmic Functions

ln cxdx = x ln cx - x

ln(ax + b)dx = x ln(ax + b) - x + b ln(ax + b) a

(ln x)2 dx = x (ln x)2 - 2x ln x + 2x

(ln cx)n dx = x (ln cx)n - n (ln cx)n-1 dx

dx = ln ln x + ln x + (ln x)i

ln x

n=2 i i!

(

dx ln x

)n

=-

x

(n -1)(ln x)n-1

+

n

1 -

1

dx

(ln x)n-1

( for n 1)

xm

ln

xdx

=

xm+1

ln x m +1

-

(m

1

+ 1)2

xm (ln x)n dx = xm+1 (ln x)n - n xm (ln x)n-1 dx

m+1 m+1

( for m 1) ( for m 1)

(ln x)n dx = (ln x)n+1

x

n +1

( for n 1)

( ) ln xn dx = ln xn 2

x

2n

( for n 0)

ln x

xm

dx

=

-

(m

ln x

-1) xm-1

-

(m

1

- 1)2

xm-1

( for m 1)

(ln x)n

xm

dx

=

-

(ln x)n (m -1) xm-1

+

n

m -1

(ln x)n-1

xm

dx

( for m 1)

dx = ln ln x x ln x

dx xn ln

x

=

ln

ln

x

+

(-1)i

i=1

(n

-1)i (ln

i i!

x)i

dx

x (ln x)n

=-

1

(n -1)(ln x)n-1

( for n 1)

( ) ( ) ln x2 + a2 dx = x ln x2 + a2 - 2x + 2a tan-1 x a

sin (ln x) dx = x (sin (ln x) - cos (ln x)) 2

cos (ln x) dx = x (sin (ln x) + cos(ln x)) 2



5. Integrals of Trig. Functions

sin xdx = - cos x

cos xdx = - sin x

sin2 xdx = x - 1 sin 2x 24

cos2 xdx = x + 1 sin 2x 24

sin3 xdx = 1 cos3 x - cos x 3

cos3 xdx = sin x - 1 sin3 x 3

dx xdx = ln tan x

sin x

2

dx cos x

xdx

=

ln

tan

x 2

+

4

dx sin2

x

xdx

=

-

cot

x

dx cos2

x

xdx

=

tan

x

dx = - cos x + 1 ln tan x

sin3 x 2sin2 x 2

2

dx

cos3

x

=

sin x 2 cos2 x

+

1 2

ln

tan

x 2

+

4

sin x cos xdx = - 1 cos 2x 4

sin2 x cos xdx = 1 sin3 x 3

sin x cos2 xdx = - 1 cos3 x 3

sin2 x cos2 xdx = x - 1 sin 4x 8 32

tan xdx = - ln cos x

sin x dx = 1 cos2 x cos x

sin2 cos

x x

dx

=

ln

tan

x 2

+

4

-

sin

x

tan2 xdx = tan x - x

cot xdx = ln sin x

cos x dx = - 1

sin2 x

sin x

cos2 x dx = ln tan x + cos x

sin x

2

cot2 xdx = - cot x - x

dx = ln tan x

sin x cos x

sin 2

dx x cos

x

=

-

1 sin

x

+

ln

tan

x 2

+

4

sin

dx x cos2

x

=

1 cos

x

+

ln

tan

x 2

sin2

dx x cos2

x

=

tan

x

-

cot

x

sinmxsinnxdx

=

sin( m+ n) - 2(m+n)

x

+

sin( m- n) 2(m-n)

x

m2 n2

sinmxcosnxdx

=

-

cos( m + n) 2( m + n)

x

-

cos ( m - n) 2( m - n)

x

m2 n2

cosmxcosnxdx

=

sin(m+ 2(m+

n) n)

x

+

sin ( m - n) 2( m - n)

x

m2 n2

sin x cosn xdx = - cosn+1 x

n +1

sinn x cos xdx = sinn+1 x

n +1

arcsin xdx = x arcsin x + 1 - x2

arccos xdx = x arccos x - 1- x2

( ) arctan xdx = x arctan x - 1 ln x2 +1 2

( ) arc cot xdx = x arc cot x + 1 ln x2 + 1 2

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