MATH141 Exam 2 Study Guide

m is odd —> u=cosx

n is odd —>  u=sinx

m,n even —> sin²x or cos²x half angle identities

n is even —> u = tanx

m is odd —> u =secx

n is odd/m is even —> write tanx in terms of secx

identity for sec²x

sec²x = tan²x + 1

tan^-1(x) + C

sin^-1(x) + C

sec^-1(x) + C

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

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

derivative of tanx

sec²x

derivative of cotx

-csc²x

derivative of secx

sec(x)tan(x)

derivative of cscx

-csc(x)cot(x)

cos²x half angle identity

(cos(2x)+1)/2

sin²x half angle identity

(cos(2x)-1)/2

integration by parts

trapezoidal rule

error for trapezoidal rule

where K_T is the second derivative of the function

Simpson’s Rule

error for simpson’s rule

where K_S is the fourth derivative of the function

comparison property (f(x) >= g(x) >= 0)

integral of f(x) converges = integral g(x) converges

integral of g(x) diverges = integral of f(x) diverges

notation for a sequence

squeeze theorem

if a <= b<= c, and lim a = lim c = L, lim b = L

series notation = limit for partial sum

formula for nth partial sum of geometric sequence

(first-after last)/(1-common ratio)

Test for divergence

diverges if not 0

Sum for Geometric Series Formula (when converging, |r|<1)

where a is the first term of the sequence