Calculus
Derivitives   Integrals   Arc Length
Curves, lines   Trigonometry   Math   Area & Volume

Derivitives

k' = 0
(kx)' = k
(f + g)' = f' + g'
(kf)' = kf'

product rule
(fg)' = f'g + fg'
(fgu)' = fgu' + fug' + guf'

chain rule
If y = f(u) and u = g(x)
dy/dx = (dy/du)(du/dx)

Easier is to break the function into the part inside and the part outside. then differentiate the outside function leaving the inside function alone and multiply all of this by the derivative of the inside function.

power rule
if f(x) = x^n
f'(x^n) = nx^(n–1)
∫ x^n dx = (1/(n+1))x^(n+1)

exponential rules
(e^x)' = e^x
(e^u)' = (e^u)u'
(x^x)' = (x^x)(1 + ln x)
(c^x)' = (c^x)ln c
(c^ax)' = (c^ax)a ln c, c>0

quotient rule
(f/g)' = (f'g–fg')/g²

log rules
(ln x)' = 1/x    x≠0
(ln ax)' = 1/x    x≠0
(x ln x)' = ln x + 1
(ln u)' = (1/x)du/dx    x≠0
   (log base k)
(log(k)x)' = 1 / (x ln k)
(log(k)u)' = (1 / (u ln k))du/dx

trig rules
(sin x)' = cos x
(sin ax)' = a cos ax
(cos x)' = –sin x
(cos ax)' = –a sin ax
(tan x)' = 1/cos²x = 1 + tan²x
(cot x)' = –csc²x = –(1+cot²x)
(sec x)' = sec x tan x
(csc x)' = –csc x cot x
(arcsin x)' = 1/√(1–x²)
(arccos x)' = –1/√(1–x²)
(arctan x)' = 1/(x²+1)
(arctan kx)' = k/(k²x²+1)

Integrals

∫ kdu = kx + C
∫ af(u) du = a∫ f(u)du
∫ [f(u)+g(u)]du = ∫ f(u)du + ∫ g(u)du
∫ [f(u)–g(u)]du = ∫ f(u)du – ∫ g(u)du
∫ [af(u)+bg(u)]du = a∫ f(u)du + b∫ g(u)du

∫ u dv = uv – ∫ v du

∫ uⁿdu = (1/(n+1))u^(n+1) + C where n ≠ –1
∫ u⁻¹du = ∫ (1/u)du = ln|u| + C
∫ e^u du = e^u + C
∫ a^u du = (1/ln a)a^u + C where a>0 and a≠1
∫ a^cu du = (1/clna)a^cu + C where a>0 and a≠1
∫ e^(au)du = (1/a)e^(au) + C
∫ ue^(au)du = (1/a²)(au–1)e^(au) + C
∫ u²e^(au)du = (1/a³)(a²u²–2au+2)e^(au) + C

∫ (1/x) dx = ln x + C
∫ ln(x)dx = x ln x – x + C
∫ ln(ax)dx = x ln ax – x + C
∫ ln(ax+b)dx = ((ax+b)ln(ax+b)–ax)/a + C
∫ (ln x)²dx = x (ln x)² – 2x ln x + 2x + C

∫ logκ x dx = (x ln x – x) / ln K + C
   (log base K)
∫ a^x dx = a^x/ln a + c

∫ sin u du = –cos u + C

∫ sin au du = –(1/a)cos au + C
∫ cos u du = sin u + C
∫ cos au du = (1/a)sin au + C
∫ sec²u du = tan u + C
∫ sec u tan u du = sec u + C
∫ csc²u du = –cot u + C
∫ csc u cot u du = –csc u + C
∫ tan u du = –ln|cos u| + C
∫ tan au du = –(1/a)ln|cos au| + C
∫ sec u du = ln|sec u + tan u| + C
∫ cot u du = –ln|sin u| + C
∫ cot au du = –(1/a)ln|sin au| + C
∫ csc u du = ln|csc u – cot u| + C

∫ sin²u du = (u/2) – (1/4)sin 2u + C

∫ sin²au du = (u/2) – (1/4a)sin 2au + C
∫ cos²u du = (u/2) + (1/4)sin 2u + C
∫ cos²au du = (u/2) + (1/4a)sin 2au + C
∫ sec³u du = (1/2)sec u tan u + (1/2)ln|sec u + tan u| + C

∫ arcsinu du = u arcsinu + √(1–u²) + C
∫ arcsin ax dx = x arcsin ax + (1/a)√(1 – a²x²) + C
∫ cos⁻¹u du = u cos⁻¹u – √(1–u²) + C
∫ tan⁻¹u du = u tan⁻¹u – (1/2)ln(u²+1) + C
∫ cot⁻¹u du = u cot⁻¹u + (1/2)ln(u²+1) + C
∫ sec⁻¹u du = u sec⁻¹u – ln(u + √(u²–1)) + C for u>1
∫ csc⁻¹u du = u csc⁻¹u + ln(u + √(u²–1)) + C for u>1

∫ 1 / (ax+b) dx = (1/a) ln |ax+b|
∫ 1 / (x–1) dx = ln |x–1|
∫ 1/(x+b)² dx = –1/(x+b)
∫ 1/(ax+b)² dx = –1/(ax+b)
∫ 1/(√(1–u²))du = arcsinu + C
∫ 1/(√(a²–u²))du = arcsin(u/a) + C for a>0
∫ 1/(√(1+u²))du = tan⁻¹u + C
∫ 1/(√(a²+u²))du = (1/a)tan⁻¹(u/a) + C for a>0
∫ 1/(u√(u²–1))du = sec⁻¹u + C for u>0
∫ 1/(u√(u²–a²))du = (1/a)sec⁻¹(u/a) + C for u>a>0

∫ 1/(a²–u²)du = (1/2a) ln [(a+u)/(a–u)] + C for u>0
∫ 1/(u²–a²)du = (1/2a) ln [(u–a)/(u+a)] + C for u>0

∫ √(a²–u²)du = (u/2)√(a²–u²) + (a²/2)arcsin(u/a) + C for a>0
∫ √(a²+u²)du = (u/2)√(a²+u²) + (a²/2) ln (u+√(a²+u²)) + C
∫ √(u²–a²)du = (u/2)√(u²–a²) – (a²/2) ln (u+√(u²–a²)) + C for u>a>0
∫ √(au+b)du = (2/3a)(au+b)^(3/2) + C
∫ u√(au+b)du = (2/15a²)(3au–2b)(au+b)^(3/2) + C


Arc Length

In rectangular coordinates:
Between x=a and x=b for runction y = f(x)
Arc length= ∫√(1+f'²)dx beteen a & b

In parametric coordinates:
Between t=a and t=b for runction y = f(t), x = g)t)
Arc length= ∫√(f'² + g'²)dt beteen a & b

In polar coordinates:
Between θ=a and θ=b for runction r = f(θ)
Arc length= ∫√(r² + f'²)dθ beteen a & b


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