Calculus 2

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sin^2(x)

(1-cox2x)/2

int sinx

-cosx+c

int csc^2(x)

-cotx+c

d/dx cotx

-csc^2(x)

int cscxcotx

-cscx + C

d/dx cscx

-cscxcotx

d/dx cosx

-sinx

cos^2(x)

1/2 + 1/2cos2x

sin^2(x)

1/2 - 1/2cos2x

int 1/(u)sqrt(u^2-a^2)

1/a[arcsec(u/a)] + C

Int 1/a^2+u^2

1/a[arctan(u/a)] +C

d/dx ln(u)

1/u • du/dx

Arc Length

L= int: sqrt ( (dy/dt)^2 + (dx/dt)^2) dy

Washer Method

V= int: piRouter^2 - piRinner^2 dx

Disk Method

V= int: pi[f(x)]^2 dx

LHopital

When lim fx and lim gx equals zero or pos/neg infinity. Lim of fx/gx equals lim f'x/g'x.

int 1/sqrt(a^2-u^2)

arcsin(u/a)+C

cos(2x)

cos^2(x) - sin^2(x)

d/dx sinx

cosx

1 + cot^2(x)

csc^2(x)

integrals of partial fractions will always work when

denominator factors into linear functions and degree of numerator is lower than degree of denominator --> if not do long division and test the Remainder

d/dx e^(f(x))

e^(f(x)) • f'(x)

d/dx e^x

e^x

int e^x dx

e^x + c

if a sereis is bounded and monotonic

it will always converge

series converges if

limit exists as a finite number

int (1/u) du

ln |u| + c

int cscx

ln|cscx-cotx| + C

int secx

ln|secx+tanx|+C

int tanx

ln|secx|+C

int cotx

ln|sinx| + C

d/dx tanx

sec^2(x)

tan^2(x) + 1

sec^2(x)

tan^2(x)+1

sec^2(x)

int secxtanx

secx+c

d/dx secx

secxtanx

if lim of an doesn't go to zero

series cannot converge

int cosx

sinx+c

int sec^2(x)

tanx + C

Integration by parts

uv - int: vdu

Trig Sub: sqrt(x^2 - a^2)

x = asec(x) and sec^2 - 1 = tan^2

Trig Sub: sqrt(a^2 - x^2)

x = asin(x) and 1 - sin^2 = cos^2

Trig Sub: sqrt( a^2+x^2)

x = atan(x) and 1 + tan^2(x) = sec^2


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