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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 n ∫ x dx =
x n +1 +C n +1
1
∫ x dx = ln x + C ∫ c dx = cx + C ∫ xdx =
x2 +C 2
x3 +C 3 1 1 ∫ x2 dx = − x + C 2 ∫ x dx =
∫
xdx = 1
∫1+ x
∫
2
2x x +C 3
dx = arctan x + C
1 1 − x2
dx = arcsin x + C
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 1 ( x − sin x cos x ) + C 2 1 2 ∫ cos x dx = 2 ( x + sin x cos x ) + C
∫ sin
2
∫ tan ∫ sec
x dx =
2
x dx = tan x − x + C
2
x dx = tan x + C
Integrals of Exponential and Logarithmic Functions
∫ ln x dx = x ln x − x + C n ∫ x ln x dx =
∫e
x
x n +1 x n +1 ln x − +C 2 n +1 ( n + 1)
dx = e x + C
x ∫ b dx =
bx +C ln b
∫ sinh x dx = cosh x + C ∫ cosh x dx = sinh x + C
www.mathportal.org 2. Integrals of Rational Functions Integrals involving ax + b
( ax + b )n + 1 ∫ ( ax + b ) dx = a ( n + 1) n
1
( for n ≠ −1)
1
∫ ax + b dx = a ln ax + b ∫ x ( ax + b )
n
a ( n + 1) x − b
dx = a
x
x
2
( n + 1)( n + 2 )
( ax + b )n+1
( for n ≠ −1, n ≠ −2 )
b
∫ ax + b dx = a − a 2 ln ax + b x
b
1
∫ ( ax + b )2 dx = a 2 ( ax + b ) + a 2 ln ax + b a (1 − n ) x − b
x
∫ ( ax + b )n dx = a 2 ( n − 1)( n − 2)( ax + b )n−1
( for n ≠ −1, n ≠ −2 )
2 x2 1 ( ax + b ) 2 dx = − 2 b ax + b + b ln ax + b ( ) ∫ ax + b 2 a3
x2
∫ ( ax + b )2 x2
∫ ( ax + b )3 x2
∫ ( ax + b ) n
1 b2 dx = 3 ax + b − 2b ln ax + b − ax + b a dx =
1 2b b2 ln ax + b + − ax + b 2 ( ax + b )2 a3
dx =
3−n 2− n 1−n 2b ( a + b ) b2 ( ax + b ) 1 ( ax + b ) − + − n−3 n−2 n −1 a3
1
1
∫ x ( ax + b ) dx = − b ln 1
ax + b x
1
a
∫ x 2 ( ax + b ) dx = − bx + b2 ln 1
∫ x 2 ( ax + b )2
ax + b x
1 1 2 ax + b dx = − a 2 + 2 − 3 ln b ( a + xb ) ab x b x
Integrals involving ax2 + bx + c 1
1
x
∫ x 2 + a 2 dx = a arctg a
a−x 1 2a ln a + x ∫ x2 − a 2 dx = 1 x − a ln 2a x + a 1
for x < a for x > a
( for n ≠ 1, 2,3)
www.mathportal.org
2 2ax + b arctan 2 4ac − b 2 4ac − b 1 2 2ax + b − b 2 − 4 ac dx = ln 2 ∫ ax 2 + bx + c b − 4ac 2 ax + b + b 2 − 4ac − 2 2ax + b x
1
∫ ax 2 + bx + c dx = 2a ln ax
2
+ bx + c −
for 4ac − b 2 > 0 for 4ac − b 2 < 0 for 4ac − b 2 = 0
b dx ∫ 2 2 a ax + bx + c
m 2an − bm 2ax + b 2 arctan for 4ac − b 2 > 0 ln ax + bx + c + 2 2 2 a a 4ac − b 4ac − b mx + n 2an − bm 2ax + b m 2 2 ∫ ax 2 + bx + c dx = 2a ln ax + bx + c + a b2 − 4ac arctanh b2 − 4ac for 4ac − b < 0 m 2an − bm ln ax 2 + bx + c − for 4ac − b 2 = 0 a ( 2 ax + b ) 2a
∫
1
( ax
∫x
2
+ bx + c
)
n
1
( ax
2
+ bx + c
)
dx =
2ax + b
( n − 1) ( 4ac − b2 )( ax 2 + bx + c )
dx =
n−1
+
( 2 n − 3 ) 2a 1 dx 2 ∫ ( n − 1) ( 4ac − b ) ( ax 2 + bx + c )n−1
1 x2 b 1 ln 2 − ∫ 2 dx 2c ax + bx + c 2c ax + bx + c
3. Integrals of Exponential Functions cx ∫ xe dx =
ecx c2
( cx − 1)
2 2x 2 2 cx cx x x e dx = e − ∫ c c 2 + c3
∫x
n cx
e dx =
1 n cx n n −1 cx x e − ∫ x e dx c c i
∞ cx ( ) ecx dx = ln x + ∑ i ⋅ i! ∫ x i =1
∫e
cx
ln xdx =
1 cx e ln x + Ei ( cx ) c
cx ∫ e sin bxdx = cx ∫ e cos bxdx = cx n ∫ e sin xdx =
ecx c 2 + b2
( c sin bx − b cos bx )
ecx c 2 + b2
( c cos bx + b sin bx )
ecx sin n −1 x 2
c +n
2
( c sin x − n cos bx ) +
n ( n − 1) 2
c +n
2
∫e
cx
sin n −2 dx
www.mathportal.org 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) 2
2
∫ ( ln x ) dx = x ( ln x ) − 2 x ln x + 2 x n n n −1 ∫ ( ln cx ) dx = x ( ln cx ) − n∫ ( ln cx ) dx i
∞ ln x ( ) dx = ln ln + ln + x x ∑ ∫ ln x n =2 i ⋅ i !
dx
∫ ( ln x )n
=−
x
( n − 1)( ln x )
n −1
+
1 dx n − 1 ∫ ( ln x )n −1
1 m m +1 ln x x l xdx x n = − ∫ m + 1 ( m + 1) 2
∫ x ( ln x ) m
∫
( ln x )n x
n
dx =
dx =
x m+1 ( ln x )
n
m +1
−
( ln x )n+1
)
( for m ≠ 1)
n n −1 x m ( ln x ) dx ∫ m +1
2
ln x n ln x n ( for n ≠ 0 ) ∫ x dx = 2n ln x ln x 1 ∫ xm dx = − ( m − 1) xm−1 − ( m − 1)2 xm−1
∫
( ln x )n xm
( for m ≠ 1)
( ln x )n ( ln x )n−1 n dx = − + dx ( m − 1) x m−1 m − 1 ∫ x m
dx
∫ x ln x = ln ln x ∞
dx
( −1) ∫ xn ln x = ln ln x + ∑ i =1 dx
∫ x ( ln x )n ∫ ln ( x
2
=−
i
( n − 1)i ( ln x )i i ⋅ i!
1
( for n ≠ 1)
( n − 1)( ln x )n−1
)
(
)
+ a 2 dx = x ln x 2 + a 2 − 2 x + 2a tan −1 x
∫ sin ( ln x ) dx = 2 ( sin ( ln x ) − cos ( ln x ) ) x
( for m ≠ 1)
( for n ≠ 1)
n +1
(
( for n ≠ 1)
∫ cos ( ln x ) dx = 2 ( sin ( ln x ) + cos ( ln x ) )
x a
( for m ≠ 1)
www.mathportal.org 5. Integrals of Trig. Functions
∫ sin xdx = − cos x ∫ cos xdx = − sin x
cos x
x 1 − sin 2 x 2 4 x 1 2 ∫ cos xdx = 2 + 4 sin 2 x 1 3 3 ∫ sin xdx = 3 cos x − cos x 1 3 3 ∫ cos xdx = sin x − 3 sin x
∫ sin
2
xdx =
dx
cos 2 x x ∫ sin x dx = ln tan 2 + cos x
∫ cot
2
xdx = − cot x − x
dx
∫ sin x cos x = ln tan x dx
dx
1
x
∫ sin 2 x cos2 x = tan x − cot x
dx
π
dx ∫ sin 2 x xdx = − cot x dx ∫ cos2 x xdx = tan x
sin( m + n) x sin( m − n) x + 2( m − n)
∫sin mxsin nxdx = − 2( m+ n)
cos ( m + n) x cos ( m − n) x − 2( m − n)
∫sin mxcos nxdx = − 2( m + n)
sin ( m + n) x sin ( m − n) x + 2( m − n)
dx cos x 1 x ∫ sin 3 x = − 2sin 2 x + 2 ln tan 2
∫ cos mxcos nxdx = 2( m + n)
dx sin x 1 x π ∫ cos3 x = 2 cos2 x + 2 ln tan 2 + 4
n ∫ sin x cos xdx = −
1 ∫ sin x cos xdx = − 4 cos 2 x 1 3 2 ∫ sin x cos xdx = 3 sin x 1 2 3 ∫ sin x cos xdx = − 3 cos x x 1 2 2 ∫ sin x cos xdx = 8 − 32 sin 4 x
n ∫ sin x cos xdx =
∫ tan xdx = − ln cos x sin x 1 dx = 2 cos x x
∫ cos
sin 2 x x π ∫ cos x dx = ln tan 2 + 4 − sin x
∫ tan xdx = tan x − x ∫ cot xdx = ln sin x
π
x
x
∫ cos x xdx = ln tan 2 + 4
2
x
1
∫ sin 2 x cos x = − sin x + ln tan 2 + 4 ∫ sin x cos2 x = cos x + ln tan 2
