Calc 2 Final

∫sec(x) dx

ln|sec(x) + tan(x)| + C

∫cos(x) dx

sin(x) + C

∫sin(x) dx

-cos(x) + C

∫sec²(x) dx

tan(x) + C

∫tan(x) dx

-ln|cos(x)| + C

∫csc²(x) dx

-cot(x) + C

∫sec(x)tan(x) dx

sec(x) + C

∫csc(x)cot(x) dx

-csc(x) + C

∫1 / √(1-x²) dx

arcsin(x) + C

∫1 / (1+x²) dx

arctan(x) + C

∫1 / |x|√(x²-1) dx

arcsec(|x|) + C

∫ln|x| dx

x ln|x| - x + C

IBP (integration by parts)

∫udv = uv - ∫vdu

sin²(x)

= ½ (1 - cos(2x))

cos²(x)

= ½ (1 + cos(2x))

sin²(x) + cos²(x)

= 1

1 + tan²(x)

= sec²(x)

1 + cot²(x)

= csc²(x)

√(a² - x²)

x = asin(θ)

√(a² + x²)

x = atan(θ)

√(x² - a²)

x = asec(θ)

If f(x)≥g(x)≥0 on the interval [a,∞) then,

If ∫a to ∞ (f(x)dx) converges then so does ∫a to ∞ g(x)dx.

If ∫a to ∞ (g(x)dx) diverges then so does ∫a to ∞ f(x)dx.

disk method

V = ∫[a to b] πf(x)² dx

washer method

V = π∫[a to b] R²-r² dx

shells method

V = ∫[a to b] 2πrf(x) dx

average value on [a, b]

1/(b-a) ∫[a to b] f(x) dx

work

∫[a to b] F(x) dx (m = pV)

average x-value x̅ (M(y)/m)

(∫[a to b] xf(x) dx) / (∫[a to b] f(x) dx)

average y-value ȳ (M(x)/m)

½ (∫[a to b] f(x)²) / ∫[a to b] f(x) dx

area under a parametric curve

A = ∫[α to β] y(t)x′(t) dt

arc length

∫[a to b] √((dx/dt)² + (dy/dt)²) dt

convert from polar to cartesian

x = rcos(θ), y = rsin(θ)

convert from cartesian to polar

θ = arctan(y/x), r = √(x² + y²)

area under a polar curve

½ ∫[θ1 to θ2] f(θ)² dθ

In a sequence, if lim(n → ∞) |aₙ| = 0, then…

lim(n → ∞) aₙ = 0

In a sequence, if lim(n → ∞) aₙ = L and f is continuous, then…

lim(n → ∞) f(aₙ) = f(lim(n → ∞) aₙ) = f(L)

A sequence {aₙ} is increasing if…

aₙ₊₁ > aₙ

A sequence {aₙ} is decreasing if…

aₙ₊₁ < aₙ

slope of a line tangent to a parametric curve

dy/dx

Divergence Test

If the limit of aₙ ≠ 0, the series ∑aₙ diverges.

Integral Test

If f(x) is pos., cont., and dec., ∑aₙ and ∫f(x)dx both converge or both diverge.

Direct Comparison Test (DCT)

If 0 ≤ aₙ ≤ bₙ and ∑bₙ converges, so does ∑aₙ. If ∑aₙ converges, so does ∑bₙ.

Limit Comparison Test

If aₙ and bₙ > 0, lim (aₙ / bₙ) = c > 0, then ∑aₙ and ∑bₙ both converge or both diverge.

Alternating Series Test

If terms alternate signs, decrease in ||, and lim = 0, ∑(–1)ⁿaₙ converges.

Ratio Test

If lim |aₙ₊₁ / aₙ|
< 1, the series converges
> 1 diverges
= 1 inconclusive

Geometric Series

∑arⁿ if |r| < 1, converges to a/(1-r), diverges if |r| ≥ 1.

P-Test

∑1/nᵖ converges if p > 1, diverges if p ≤ 1.

Telescoping Series

Terms cancel out; use partial sums to find convergence.

Alternating Series Remainder

|Rₙ| ≤ |aₙ₊₁|

Integral Remainder

∫ [n+1,∞] f(x) dx ≤ Rₙ ≤ ∫ [n,∞] f(x) dx

Taylor Series (∑ notation)

f(x) = ∑ (f⁽ⁿ⁾(a)/n!) · (x – a)ⁿ

Taylor Series (expanded form)

f(x) = f(a) + f′(a)(x–a) + f″(a)(x–a)²/2! + ... + fⁿ(a)(x–a)ⁿ/n!

e^x Taylor Series

∑(0 to ∞) xⁿ / n! (RoC: ∞)

sin(x) Taylor Series

∑(0 to ∞) (-1)ⁿx^(2n+1) / (2n+1)! (RoC: ∞)

cos(x) Taylor Series

∑(0 to ∞) (-1)ⁿx^(2n) / (2n)! (RoC: ∞)

1/(1-x) Taylor Series

∑(0 to ∞) xⁿ (RoC: 1)

arctan(x) Taylor Series

∑(0 to ∞) (-1)ⁿx^(2n+1) / 2n+1 (RoC: ∞)

ln(1+x) Taylor Series

∑(1 to ∞) (-1)^(n-1)xⁿ / n (RoC: ∞)

|Rn(x)| <=

M/(n+1)! |x-a|^(n+1)