Calc II - Exam 1

sin(2x)

2sinxcosx

cos(2x)

cos²x-sin²x

cos(2x)

2cos²x-1

cos(2x)

1-2sin²x

tan(2x)

2tanx/1-tan²x

sin(x/2)

+-√1-cosx/2

cos(x/2)

+-√1+cosx/2

tanx/2

+-√1-cosx/1+cosx

Product Rule

f'(x)g(x)+f(x)g'(x)

Quotient Rule

g(x)f'(x)-f(x)g'(x)/g(x)^2

Chain Rule

f'[g(x)]g'(x)

Derivative of sin(x)

cos(x)

Derivative of cos(x)

-sin(x)

Derivative of tan(x)

(secx)^2

Derivative of sec(x)

sec(x)tan(x)

Derivative of csc(X)

-csc(x)cot(x)

Derivative of cot(x)

-(cscx)^2

Derivative of arcsin(x)

1/√(1-x^2)

Derivative of arccos(x)

- 1/√(1-x^2)

Derivative of arctan(X)

1/(1+x^2)

Derivative of arcsec(x)

1/(|x|√(x)^2 -1)

Derivative of arccsc(x)

1/(|x|√(x)^2 -1)

Derivative of arccot(x)

-1/(1+x^2)

Derivative of (a)^x

(a)^x ln(a)

Derivative of e^x

e^x

Derivative of ln(x)

1/x where x>0

Derivative of ln|x|

1/x where x≠0

(logₐ(x))

1/[xln(a)] , x > o

Intergral of cos(x)dx

sin(x)+c

Intergral of sin(x)dx

-cos(x)+c

Intergral of [(secx)^2]dx

tan(x)+c

Intergral of sec(x)tan(x)dx

sec(x)+c

Intergral of csc(x)cot(x)dx

-csc(x)+c

Intergral of [(cscx)^2]dx

-cot(x)+c

Intergral of tan(x)dx

ln|sec(x)|+c

Intergral of cot(x)dx

ln|sin(x)|+c

Intergral of sec(x)dx

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

Intergral of ([sec(x)]^3) dx

(1/2)(sec(x)tan(x) + ln|sec(x)+tan(x)|)+c

Intergral of csc(x)dx

ln|csc(x) - cot(x)|+c

Intergral of ([csc(x)]^3)dx

(1/2)(-csc(x)cot(x)+ln|csc(x)-cot(x)|)+c

Derivative arcsin u

U' / sqrt(1-u^2)

Derivative arctan u

U'/(u^2+1)

Derivative arcsec u

U' / abs(u)sqrt(u^2-1)

Derivative tan x

Sec ^2 x

Derivative log base a of u

U' /( ln a *u)

Drivative sec x

Sec x tan x

∫ du / sqrt a^2 - u^2

Arcsin (u/a)

∫ du / a^2 + u^2

(1/a)arctan(u/a)

∫ du/|u|sqrt(u^2-a^2)

1/a arcsec(|u|/a)

∫ tan x

-ln|cos x|

∫sec x dx

Ln|secx +tanx|

∫cot x dx

Ln|sinx|

Integration by parts

∫udv = vu - ∫vdu

Integrating sin power times cos power

If cos power is odd, u = sin

If sin power is odd, u = cos

If neither, use half angle formula sin^2 = 1-cos(2x) over 2

Integrating sec power times tan power

If sec is even, use tan

If tan is odd, use sec

If sec is even but tan is 0, integral by parts

Else cos and sin

√a^2-x^2 substitute

a sin u = x

√x^2-a^2 substitute

a sec u = x

√x^2 + a^2 substitute

a tan u = x