Calculus 2 Review

d/dx cos⁻¹ x

-1/√(1 - x²)

d/dx tan⁻¹ x

1/(1 + x²)

d/dx log(a)x

1/(x ln a)

d/dx ln |x|

1/x

d/dx sin⁻¹ x

1/√(1 - x²)

What is the half angle formula for sin 2∅?

2 sin ∅ cos ∅

What is l'Hospital's rule?

If (lim x→a f(x) = 0 and lim x→a g(x) = 0) or (lim x→a f(x) = ±∞ and lim x→a g(x) = ±∞) then lim x→a f(x)/g(x) = lim x→a f'(x)/g'(x).

∫ aⁿ dn

aⁿ / ln a + c

d/dx aⁿ

aⁿ ln a

What is the double angle formula for cos 2∅?

cos² ∅ - sin² ∅ 2cos² ∅ - 1 1 - 2 sin² ∅

d/dx eⁿ

eⁿ dn/dx

∫ sec x dx

ln | sec x + tan x | + c

∫ tan x dx

ln | sec x | + c

∫1/x

ln |x| + c

What is the change of base formula?

log(a)x = (ln x) / (ln a)

What is the Pythagorean identity for tan² ∅?

sec² ∅ - 1

∫ 1/√(1-x²) dx

sin⁻¹ x + c

What is the Pythagorean identity for sec² ∅?

tan² ∅ + 1

∫ 1/(x²+1) dx

tan⁻¹ x + c

What is the trigonometric substitution for √(x² - a²)?

x = a sec ∅ for 0 < ∅ < π/2 or π < ∅ < 3π/2

What is the trigonometric substitution for √(a² - x²)?

x = a sin ∅ for -π/2 ≤ ∅ ≤ π/2

What is the trigonometric substitution for √(a² + x²)?

x = a tan ∅ for -π/2 < ∅ < π/2

What is the double angle formula for cos² ∅?

½(1 + cos 2∅)

What is the double angle formula for sin² ∅?

½(1 - cos 2∅)

What is the formula for integration by parts?

∫ u dv = uv - ∫ v du

What is the formula for the surface area of the surface swept by rotating a parameterized curve about the y axis?

∫(a->b)2πf(x)√( (dx/dt)² + (dy/dt)²)dx

What is the formula for the surface area of the surface swept by rotating a parameterized curve about the x axis?

∫(a->b)2πg(x)√( (dx/dt)² + (dy/dt)²)dx

What is the formula for the surface area of the surface swept by rotating a curve about the y axis?

∫(a->b)2πx√(1 + (dx/dy)²)dx

What is the formula for the surface area of the surface swept by rotating a curve about the x axis?

∫(a->b)2πy√(1 + (dx/dy)²)dx

What is the formula for the length of a parameterized curve?

∫(a->b)√( (dx/dt)² + (dy/dt)²)dx

What is the formula for arc length?

∫(a->b)√(1 + (dx/dy)²)dx