f'(sinx)
cosx
int(-sinx)
-cosx+c
int(sec^2(x))
tan(x)+C
int (secxtanx)
secx+C
f'(cscx)
-cscxcotx
f'(cotx)
-csc^2(x)
Geometric series
r>1=diverge, r<1=converge
Absolute convergence test
abs(An) converges, then An converges absolutely
Alternating series test
1). Un>0 2). Un>Un+1 3). lim (Un) = 0
P series test
p>1=convergence, 0<p<1= divergence
Ratio Test
lim n→∞ abs ( An+1/An ) converges if L < 1, diverges if L > 1, inconclusive if L = 1
Root Test
lim n-> infinity |a_n|^(1/n) = R, then R<1 converges, R>1 diverges, R=1 no conclusion
Washer Method
Outer Disk - Inner Disk
Shell Method
∫ 2πr*r*f(x) (opposite)
Disk Method
integral{ pi[f(x)]^2 dx
Integration by Parts
∫u dv=uv-∫v du
U substitution
u=x du=dx, integrate
fundamental theorem of calculus
∫f(x) dx=F(b)-F(a)
sqrt (a^2+x^2)
asec(0)
sqrt(a^2-x^2)
acos(0)
sqrt(x^2-a^2)
atan(0)
Partial Fractions
1/(x+n)(x+m) = A/(x+n) + B/(x+m)
Limit Comparison test
an, bn positive, if lim an/bn =c, c>0 converges if bn converges; finite and positive both converge or diverge, if limit is 0 or DNE inconclusive
Power series
the sum of terms containing successively higher integral powers of a variable
sin^2(u)
1-cos^2(u)
1+tan^2(u)
sec^2(u)
1+cot^2(u)
csc^2(u)
sin(2u)
2sinucosu
cos(2u)
cos^2u-sin^2u 2cos^2u-1 1-2sin^2
tan(2u)
(2tanu)/(1-tan²u)
Taylor Series
Series from n=0 to infinity of c_n (x-a)^n, where c_n= (f^n(a))/(n!)
lim ln(n)/n
0
lim n(root(n))
1
lim x^(1/n)
1
lim x^n
0 x<1
lim(1+x/n)^n
e^x
lim x^n/n!
0
int (sec x dx)
ln abs(sec x + tan x) +C
int (secxtanx)
secx +C
1/(1+x^2)
tan^-1(x) + C
Taylor Polynomial
f(a)+f'(a)/1!*(x-a)^1+(f"(a)/2!)(x-a)...