CALC 2 - Trig sub to partial fractions

sin(x) derivative

cos(x)

cos(x) derivative

-sin(x)

tan(x) derivative

sec²(x)

cot(x) derivative

-csc²(x)

sec(x) derivative

sec(x)tan(x)

csc(x) derivative

-csc(x)cot(x)

∫sin(x) dx

-cos(x) + C

∫cos(x) dx

sin(x) + C

∫tan(x) dx

-ln|cos(x)| + C

∫cot(x) dx

ln|sin(x)| + C

∫sec(x) dx

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

∫csc(x) dx

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

∫sec³(x) dx

½(sec(x)tan(x)+ln|sec(x)+tan(x)|)+C

∫csc³(x) dx

-½(csc(x)cot(x)+ln|csc(x)+cot(x)|)+C

arcsin(x) derivative

1/√(1-x²)

arccos(x) derivative

-1/√(1-x²)

arctan(x) derivative

1/(1+x²)

arccot(x) derivative

-1/(1+x²)

arcsec(x) derivative

1/(|x|√(x²-1))

arccsc(x) derivative

-1/(|x|√(x²-1))

∫dx/√(1-x²)

arcsin(x)+C

∫-dx/√(1-x²)

arccos(x)+C

∫dx/(1+x²)

arctan(x)+C

∫-dx/(1+x²)

arccot(x)+C

∫dx/(x√(x²-1))

arcsec(x)+C

∫-dx/(x√(x²-1))

arccsc(x)+C

sin power odd strategy

save one sin(x) then use u=cos(x)

cos power odd strategy

save one cos(x) then use u=sin(x)

sin and cos both even strategy

use half-angle identities

sec power even strategy

save sec²(x) then use u=tan(x)

sec and tan both odd strategy

save sec(x)tan(x) then use u=sec(x)

sec odd tan even strategy

convert tangents to secants and expand

cot power odd strategy

save csc(x)cot(x) then use u=csc(x)

csc power even strategy

save csc²(x) then use u=cot(x)

∫csc(x) dx

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

∫csc³(x) dx

-½(csc(x)cot(x)+ln|csc(x)+cot(x)|)+C

trig sub √(a²-u²)

u=a·sinθ → √(a²-u²)=a·cosθ

trig sub √(a²+u²)

u=a·tanθ → √(a²+u²)=a·secθ

trig sub √(u²-a²)

u=a·secθ → √(u²-a²)=a·tanθ

quadratic ax²+bx+c integral

complete the square then trig sub

∫dx/(a²+x²)

(1/a)arctan(x/a)+C

partial fractions distinct linear

A/(ax+b)+B/(cx+d)+…

partial fractions repeated linear

A₁/(ax+b)+A₂/(ax+b)²+…+Aₙ/(ax+b)ⁿ

partial fractions irreducible quadratic

(Ax+B)/(ax²+bx+c)

partial fractions repeated irreducible quadratic

(A₁x+B₁)/(ax²+bx+c)+…+(Aₙx+Bₙ)/(ax²+bx+c)ⁿ