PDF -Integral Calculus Formula Sheet - Calc Formulas

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## Calc Formulas

### Integral Calculus Formula Sheet

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# Description

### Formulas from Calculus Derivatives [ex ]

• = bx ln b
• d dx d'dx d'dx
• = − x12
• [logb x]
• d dx d'dx
• = − x23
• d dx d'dx
• [sinh x]
• = cosh x
• [cosh x]
• = sinh x
• d dx d'dx d'dx
• [tanh x]
• = sech 2 x
• [arcsinh x]
• d dx d'dx d'dx
• = nxn−1
• √ [ x] √1 x

1 √ 2 x

• = − 2x1√x
• [arctanh x] =
• [sin x]
• = cos x
• [cos x]
• = − sin x
• d dx d'dx d'dx
• [tan x]
• = sec2 x

1 x ln b

• [sec x]
• = tan x sec x
• d dx d'dx
• [arcsin x]
• √ 1 1+x2 1 1−x2

[f (x)g(x)] = f 0 (x)g(x) + f (x)g 0 (x)

# Product Rule:

### Quotient Rule:

d f (x) g(x)f 0 (x) − f (x)g 0 (x) = dx g(x) g(x)2

Chain Rule:

d [f (g(x))] = f 0 (g(x))g 0 (x) dx

# Special Cases d'[f (x)n ] dx d'1 dx g(x) 

d [ln |f (x)|] = dx d'h f (x) i e dx

• = nf (x)n−1 f 0 (x)
• [arctan x] =

−g 0 (x) g(x)2 f 0 (x) f (x)

• = f 0 (x)ef (x)
• dy dy du = dx du dx
• √ 1 1−x2 1 1+x2

Integrals Z

• 1 dx Z x c'dx Z
• sinh x dx
• = cosh x + C
• 1 2 2x 1 3 3x
• x ln x dx
• xn+1 n+1
• = − 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
• − sin x cos x) + C
• cos2 x dx
• + sin x cos x) + C
• ln x −
• xn+1 (n+1)2
• tan2 x dx = tan x − x + C
• sec2 x dx
• f (g(x))g (x) dx =

f (u) du = F (u) + C = F (g(x)) + C Z g(b)

• f (u) du
• f (g(x))g (x) dx = g(a)

f 0 (x) dx = ln |f (x)| + C f (x)

ef (x) f 0 (x) dx = ef (x) + C

By parts

• u dv = uv −
• u dv = a
• v du −

f (x)g (x)dx = f (x)g(x) − 0

• = tan x + C
• sin x dx Z
• 1 √ dx x Z 1 dx = arctan x + C 1 + x2 Z 1 √ dx = arcsin x + C 1 − x2 Z ln x dx = x ln x − x + C

1 x ln b b

• cosh x dx = sinh x + C
• f (x)g (x)dx = a
• 1 +C x √ = 23 x x + C √ = 2 x+C
• 1 dx 2 Z x √ x dx Z

Special cases

• = ex + C
• = cx + C

### Substitution

• = ln |x| + C
• 1 n+1 n+1 x
• f (x)g(x)]ba
• g(x)f 0 (x) dx Z b a
• g(x)f 0 (x) dx