Integration Formulas
[Pages:5]
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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