Review of Limits, Derivatives, and Integrals

lim x→∞ex

∞.

lim x→∞e−x

0

lim x→∞xa

∞ (for any a > 0).

lim x→∞√x

∞ since √x = x^(1/2) which corresponds to a = 1/2 > 0.

lim x→∞1/xa

0 (for any a > 0).

lim x→∞ln(x)

∞.

lim x→∞arctan(x)

π/2.

lim x→−∞arctan(x)

−π/2.

lim x→∞sin(x)

doesn't exist.

lim x→−∞sin(x)

doesn't exist.

lim x→∞cos(x)

doesn't exist.

lim x→−∞cos(x)

doesn't exist.

lim x→0+1/xa

∞ (for a > 0).

lim x→0+ln(x)

−∞.

d/dxf(x) ± g(x)

f ′(x) ± g′(x) (Sum and Difference Rules).

d/dxf(x)g(x)

f ′(x)g(x) + f(x)g′(x) (Product rule).

d/dx f(x)/g(x)

f ′(x)g(x) − f(x)g′(x) / (g(x))^2 (Quotient Rule).

d/dxf(g(x))

f ′(g(x))g′(x) (Chain Rule).

d/dxc

0 for any constant c.

d/dxex

ex.

d/dxxn

nxn−1.

d/dxln(x)

1/x.

d/dxsin(x)

cos(x).

d/dxcos(x)

−sin(x).

d/dxarctan(x)

1/(1 + x^2).

d/dxarcsin(x)

1/√(1 − x^2).

d/dxax

ax ln(a).

d/dxtan(x)

sec^2(x).

d/dxsec(x)

sec(x) tan(x).

∫x^ndx

1/(n + 1)x^(n+1) + C if n ≠ −1.

∫exdx

ex + C.

∫1/xdx

ln(|x|) + C.

∫sin(x)dx

−cos(x) + C.

∫cos(x)dx

sin(x) + C.

∫sec^2(x)dx

tan(x) + C.

∫sec(x)tan(x)dx

sec(x) + C.

∫tan(x)dx

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

∫sec(x)dx

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

∫1/(1 + x^2)dx

arctan(x) + C.

∫1/√(1 − x^2)dx

arcsin(x) + C.

sin^2(ω) + cos^2(ω)

1

sin(2ω)

2 sin(ω) cos(ω).

sec^2(ω)

1 + tan^2(ω).

tan(ω)

sin(ω)/cos(ω).

sec(ω)

1/cos(ω).

csc(ω)

1/sin(ω).

cot(ω)

cos(ω)/sin(ω).

tan(ω) in a right triangle

opposite side over adjacent side.

sin(ω) in a right triangle

opposite side over hypotenuse.

cos(ω) in a right triangle

adjacent side over hypotenuse.

sin^2(ω)

1-cos^2(ω) or 1-cos(2ω)/2

cos^2(ω)

1-sin^2(ω) or 1+cos(2ω)/2

sinαcosβ

=1/2[sin(α−β)+sin(α+β)]

sinαsinβ

1/2[cos(α−β)−cos(α+β)]

cosαcosβ

1/2[cos(α−β)+cos(α+β)]

sin(ω)cos(ω)

sin(2ω)/2

trig integrals: ∫sin^m*cos^n

- m odd: extract a sin, convert the rest to cos, u sub cos

- n odd: extract a cos, convert the rest to sin, u sub sin

- neither is odd: half angle or double angle formulas

trig integrals: ∫sec^m*tan^n

- m even, >2: extract 2 sec, convert rest to tan, u sub tan = sec^2

- n odd, there are some sec: extract one of each, conv. tan -> sec, u sub sec = sectan

- else rewrite into sin, cos.... tan = sin/cos, cot = cos/sin

trig substitution cases

sqrt(a²-x²) → x = a sin(θ)

sqrt(a²+x²) → x = a tan(θ)

sqrt(x²-a²) → x = a sec(θ)