Integral rules

k

kx

x^n (n =/= -1)

x^n+1 / n + 1

1/x

ln(x)

sin(ax)

-cos(ax) / a

cos(ax)

sin(ax) / a

sec(x)tan(x)

sec(x)

csc(x)cot(x)

-csc(x)

sec²(x)

tan(x)

csc²(x)

-cot(x)

e^ax

e^ax / a

a^x (a>1)

1 / ln(a) * a^x

1 / 1+x²

arctan(x)

1 / sqrt(1- x²)

arcsin(x)

derivative limit notation

lim as h → 0 = (f(x+h) - f(x)) / h

related rates steps

1. find rate you want to solve for

2. set up equation to solve for rate

3. substitute and solve

fundamental theorem

integral from a to b f(x) dx = F(b) - F(a)

F is antiderivative

substitutions u/v

1. find derivative of u with respect to x

2. solve for dx

3. change bounds of interval according to substitution (plug in values)

4. substitute into integral

tangent line approximation

y-y0 = m(x-x0)

avg value of integral

f_avg = 1/b-a * integral(a to b) f(x) dx

1. find zeroes/a and b values

2. find antiderivative

3. plug into equation

linear approximation

L(x) = f(a) + f’(a) (x-a)

integration by parts eqn

integral u dv = uv - integral v dv

total change theorem

f(b) - f(a) = integral a to b of f ’(x) dx

net signed area

solve integral from a to x of f(t) dt

optimization

draw picture
identify quantity to be optimized and find relationships
solve for a single variable

go on to solve for that variable and use it to solve for the others

1st deriv test

if f’ changes to negative you have maximum

if f’ changes to positive you have minimum

2nd deriv test

f’’(p) < 0, relative max

f’’(p) > 0, relative min

implicit differentiation

differentiate both sides with repsect to x, treating y as a differentiable function of x

collect terms with dy/dx on one side, solve for dy/dx

deriv of log

log_a (x) = 1/x*ln(a)