Mathematics reference
Rules for integration
18Ma6
MathRef
Essential rules for integration.

Legend.
  • a and n are scalars,
  • u and v are functions of x,
  • e is the base of the natural logarithm.
  • d is the differential operator,
  • int represents the integral symbol.
  • C represents the constant of integration,
Basic.
int 0 dx = C equation 1
int a u dx = a int u dx + C equation 2
int (u + v) dx = int u dx + int v dx + C equation 3
int u dv = u v - int v du equation 4
int dx = x + C equation 5
int xn dx = xn + 1/(n + 1) + C, n != -1 equation 6
int dx/x = ln |x| + C equation 7
int ex dx = ex + C equation 8
int (du/dx) dx = u + C equation 9
(d/dx) int u dx = u equation 10
Trigonometry.
int sin x dx = -cos x + C equation 11
int cos x dx = sin x + C equation 12
int tan x dx = -ln |cos x| + C equation 13
int cot x dx = ln |sin x| + C equation 14
int sec x dx = ln |sec x + tan x| + C equation 15
int csc x dx = -ln |csc x + cot x| + C equation 16
int sec2 x dx = tan x + C equation 17
int sec x tan x dx = sec x + C equation 18
int csc2 x dx = -cot x + C equation 19
int csc x cot x dx = -csc x + C equation 20
int dx/(a2 - x2)1/2 = arcsin (x/a) + C equation 21
int dx/(a2 + x2) = (1/a) arctan (x/a) + C equation 22
int dx/[x (x2 - a2)1/2] = (1/a) arcsec (|x|/a) + C equation 23
Hyperbolic trigonometry.
int cosh x dx = sinh x + C equation 24
int sinh x dx = cosh x + C equation 25
int sech2 x dx = tanh x + C equation 26
int csch2 x dx = -coth x + C equation 27
int sech x tanh x dx = -sech x + C equation 28
int csch x coth x dx = -csch x + C equation 29
int dx/(x2 + 1)1/2 = arcsinh x + C equation 30
int dx/(x2 - 1)1/2 = arccosh x + C equation 31
int dx/(1 - x2) = arctanh x + C, |x| < 1 equation 32
int dx/(1 - x2) = arccoth x + C, |x| < 1 equation 33
int dx/[x (1 - x2)1/2] = -arcsech x + C equation 34
int dx/[|x| (1 + x2)1/2] = -arccsch x + C equation 35
Other.
int dx/(x2 +- a2)1/2 = ln [x + (x2 +- a2)1/2] + C equation 36
int dx/(a2 - x2) = [1/(2 a)] ln |(a + x)/(a - x)| + C equation 37
int dx/[x (a2 +- x2)1/2] = (1/a) ln [[a + (a2 +- x2)1/2]/|x|] + C equation 38
int (a2 - x2)1/2 dx = (1/2) [a2 arcsin (x/a) + x (a2 - x2)1/2] + C equation 39
int (x2 +- a2)1/2 dx = (1/2) [x (x2 +- a2)1/2 +- a2 ln |x + (x2 +- a2)1/2|] + C equation 40
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