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