Integration Formulas

x

∫ x

∫

∫

∫

∫

w ww . m a t h p or t a l . o r g Integration Formulas1. Common IntegralsIndefinite Integral

Integrals of Exponential and Logarithmic Functions

∫ ln x dx = x ln x − x + CMethod of substitution x n +1

xn x dx = x n +1

x − + C∫ f ( g ( x)) g ′( x)dx = ∫ f (u)du∫ ln

lnn + 1

( n +1)2

Integration by parts∫ ex dx = ex + C

∫ f ( x) g ′( x)dx =

f ( x) g ( x) − ∫ g ( x) f ′( x)dx b x dx =

b + C

Integrals of Rational and Irrational Functionsn +1

∫ ln b

xn dx = x

+ Cn + 1 ∫ sinh x dx = cosh x + C

∫ cosh x dx = sinh x + C1∫ xdx = ln x + C

∫ c dx = cx + C

x2

∫ xdx = 2

+ C

3

x2 dx = + C3

1 1∫ dx = − + C

x2 x

xdx = 2 x x

+ C3

1 dx = arctan x + C∫ 1 + x2

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

sin 2 x dx = 1

( x − sin x cos x ) +

C2

cos2 x dx = 1

( x + sin x cos x ) +

C2

w ww . m a t h p or t a l . o r g

∫ tan 2 x dx = tan x − x + C

∫ sec2 x dx = tan x + C

∫

∫

∫

∫

∫

∫

∫


2. Integrals of Rational FunctionsIntegrals involving ax + b

+ n + 1

( ax + b )n

dx = ( ax b )

a ( n + 1) 1

dx = 1

ln ax + b ax + b a

( for n ≠ −1)

x ( ax + b )n

dx = a ( n + 1) x − b

( ax +

b)n+1

a2 ( n + 1) ( n + 2) x

dx = x

− b

ln ax + b ax + b a a2

x dx =

b +

1 ln ax + b

( for n ≠ −1, n ≠ −2)

∫ ( ax + b

)2

a2 ( ax + b) a2

x dx =

a (1 − n ) x − b ( for n ≠ −1, n ≠ −2)

( ax + b )n

a2 ( n − 1) ( n − 2) ( ax + b)n−1

2 + 2 x

dx = 1

( a x b ) − 2b ( ax + b ) + b2 ln ax + b

ax + b a3 2

x2 1 b2

∫ ( ax + b

)2

a3

ax b

dx = ax + b − 2b ln ax + b − +

x2 1 2b b2 ∫ dx = ln ax + b + −

( ax + b )3 a3 ax + b 2 ( ax + b )

2

2 +3−n

+ 2−n 2 +

1−n x dx =

1 − ( ax b )

+ 2b ( a b )

− b ( ax b )

( for n ≠ 1, 2, 3)

( ax + b )n a3 n − 3 n − 2 n − 1

1

dx = − 1

ln ax + b∫ x ( ax + b ) b x

1 dx = −

1 +

a ln

a x + b∫ x2 ( ax + b )

bx b2 x

1 1 1 2 a x + b ∫ dx = −a + − ln

x2 ( ax + b )2 b2 ( a + xb ) ab2 x b3 x

Integrals involving ax2 + bx + c

1 dx =

1 arctg

x∫ x2 + a2 a a 1

ln a − x

for x < a 1

dx = 2a a + x∫ x2 − a2 1

ln x − a

for x > a 2a x + a

2

∫

cx

∫ ∫

dx


2 arctan

2 ax + b for 4ac − b2 > 0

4ac − b2 4ac − b2

+ − 2 − 1 2 2 ax b b 4 ac

∫ dx = ln for 4ac − b2 < 0ax2 + bx + c b2 − 4ac

2ax + b + b2 − 4ac

− 2

for 4ac − b2 = 0 2ax + b

x dx =

1 ln ax2

+ bx + c − b dx ∫ ax2 + bx + c 2a 2a ∫ ax2 + bx + c

m ln ax2 + bx + c +

2 a n − b m arctan

2 ax + b for 4ac − b2 > 0

2a a 4ac − b2 4ac − b2

+ − + mx n∫ dx = m

ln ax2 + bx + c +2an bm arctanh 2ax b

for 4ac − b2 < 0ax2 + bx + c 2a a b2 − 4ac b2 − 4ac

m 2 a n − bm ln ax + bx + c − for 4ac − b2 = 02a a ( 2ax + b )

1 2ax + b ( 2n − 3) 2a 1∫ (ax2 + bx + c )

ndx = +( n − 1) ( 4ac − b2 ) ( ax2 + bx + c

)n−1

( n − 1) ( 4ac − b2 ) ∫ (ax2 + bx + c )n−1

1 1 x2 b 1x ( ax2 + bx + c

)

dx = ln2c ax2 + bx + c

− 2c ∫ ax2 + bx + c

dx

3. Integrals of Exponential Functions

cx e ∫ xe dx = c2 (cx − 1)

2

x2 ecx dx = ecx x −

2 x +

2 ∫ c c2 c3

xn ecx dx = 1

xn ecx − n

xn−1ecx

dx c c

ecx

∫ x

∞

dx = ln x + ∑i=1

(cx )i

i ⋅ i !

ecx ln xdx = 1

ecx ln x + E ( cx )∫ c i

cxcx e ∫ e sin bxdx

=2 2 (c sin bx − b cos bx )

c + bcx

cx e ∫ e cos bxdx

=2 2 (c cos bx + b sin bx )

c + b

( )


∫ ecx sin n

xdx = e cx sin n −1 x c sin x − n cos bx +

c2 + n2

n ( n − 1 ) c2 + n2

∫ ecx sin n−2 dx

a

∫

∫


4. Integrals of Logarithmic Functions

∫ ln cxdx = x ln cx − x

b∫ ln(ax + b)dx = x ln(ax + b) − x + a

ln(ax + b)

∫ ( ln x )2

dx = x (ln x )2

− 2 x ln x + 2x

∫ ( ln cx )n

dx = x (ln cx )n

− n∫ (ln cx )n−1

dx

dx∫ ln x

∞= ln ln x + ln x + ∑

n=2

(ln x )i

i ⋅ i !

dx x 1 d x ∫ = − + ∫ ( for n ≠ 1)

( ln x )n

( n − 1) ( ln x )n−1

n − 1 ( ln x )n−1

∫ xm ln xdx = xm+1 ln x −

1 ( for m ≠ 1) m + 1 ( m + 1)2

m+1 n

xm (ln x )n dx = x ( ln x )

− n

xm (ln x )n−1 dx ( for m ≠ 1)∫ m + 1 m + 1 ∫

( ln x ) n

( ln x ) n+1

dx = ( for n ≠ 1)x n + 1

(ln xn )2

ln x n∫ x dx =

2n ( for n ≠ 0)

ln x dx = −

ln x −

1 for m ≠

∫ m m−1 2m−1 ( 1)

x ( m − 1) x ( m − 1) x

( ln x ) n

( ln x ) n

n ( ln x ) n−1

∫ dx = − + ∫ dx ( for m ≠ 1)xm ( m − 1) xm−1

m − 1 xm

dx ∫ x ln x = ln ln x

∞ i i dx i ( n − 1) ( ln x ) ∫ = ln ln x + ∑ ( −1)

xn ln x

dx∫ n = −

i=1

1n−1

i ⋅ i !

( for n ≠ 1)x ( ln x ) ( n − 1) (ln x )

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

sin ( ln x ) dx = x

(sin (ln x ) − cos (ln x ))2

∫w ww . m a t h p or t a l . o r g

cos (ln x ) dx = x

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

∫

1

∫ 3 3

∫ 2 3

1

∫∫

∫

∫


5. Integrals of Trig. Functions

∫ sin xdx = − cos x c os x dx = −

1 sin2 x sin x

∫ cos xdx = − sin x

co s 2 x x∫ dx = ln tan + cos x

∫ sin 2 xdx = x

− 1

sin 2x

sin x 2

2 4

∫ cos2 xdx = x

+ 1

sin 2 x

∫ cot 2 xdx = − cot x − x

2 4

∫ sin3 xdx = cos3 x − cos x

dx∫ sin x cos x= ln tan x

3 dx = −

1 + ln tan x

+ π

1cos xdx = sin x − sin x3

∫ sin 2

∫

x cos x

dx2

sin x

= 1

+ ln tan

2 4

x dx x

∫ xdx = ln tan sin x cos x cos x 2sin x 2 dx

∫ 2 2= tan x − cot x

d x xdx = ln tan

x + π

sin x cos xs i n ( m + n ) x

s i n ( m − n ) x

cos x

dx

2 4 ∫sinmxsinnxdx = −

2(m+ n) + 2(m− n)m2 ≠ n2

∫ sin2 x xdx = − cot x

( ) ( ) d x

cos m+ n x cos m− n x

∫sin mxcosnxdx = − −m2 ≠ n2

∫ cos2 x xdx = tan x 2(m+ n) 2(m− n)

sin ( m + n) x sin ( m − n) x

∫cosmxcosnxdx = +m2 ≠ n2

dx∫ sin3 x= −

cos x2 sin 2

x

+ 1

ln tan x

2 2

2(m+ n)cosn +1 x

2(m− n)

d x =

s in x +

1 ln tan x

+ π

∫ sin x cosn xdx = − n + 1

cos3 x 2 cos2 x 2 2 4 sinn +1 x1∫ sin x cos xdx = − 4

cos

2x∫ sin n x cos xdx

=n + 1

1sin x cos xdx = sin x3 ∫ arcsin xdx = x arcsin x

+

1 − x2

∫ sin x cos2 xdx = − cos3

x

∫ arccos xdx = x arccos x

−

1 − x2

3

sin 2 x cos2 xdx = x

− 1

sin

4x8 32

∫ tan xdx = − ln cos x

s in x dx =

1

∫ ( 2 )∫ ( 2 )w ww . m a t h p or t a l . o r g

1arctan xdx = x arctan x − ln x + 1

2

1arc cot xdx = x arc cot x + ln x + 12

cos2 x

sin2 x

cos x

x π

∫ cos x 2 4

dx = ln tan +

∫ tan 2 xdx = tan x − x

∫ cot xdx = ln sin x

− sin x

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Integration Formulas