integration + differentiation

∫ ke^kx dx

∫ ke^kx dx = ke^kx + c

∫ - ke^kx dx

∫ - ke^kx dx = - ke^kx + c

∫ x^-1 dx

∫ x^-1 dx = ln|x| + c

∫ cosx dx

∫ cosx dx = sinx + c

∫ sinx dx

∫ sinx dx = - cosx + c

∫ - cosx dx

∫ - cosx dx = - sinx + c

∫ - sinx dx

∫ - sinx dx = cosx + c

∫ sec²x dx

∫ sec²x dx = tanx + c

∫ - sec²x dx

∫ - sec²x dx = - tanx + c

∫ cosec²x dx

∫ cosec²x dx = - cotx + c

∫ - cosec²x dx

∫ cosec²x dx = cotx + c

∫ cosecxcotx dx

∫ cosecxcotx dx = - cosecx + c

∫ - cosecxcotx dx

∫ - cosecxcotx dx = cosecx + c

∫ secxtanx dx

∫ secxtanx dx = secx + c

∫ - secxtanx dx

∫ - secxtanx dx = - secx + c

trapezium rule;

∫^b_a y dx ≈ ?

∫^b_a y dx ≈ ½ h (y₀ + 2 (y₁ + y₂ + … + y_n-1) + y_n)

• h = (b - a) / n

• n = number of columns (number of strips -1)

• y_i = f (a + ih)

what is h in the trapezium rule?

h = (b - a) / n

• ∫^b_a y dx

• n = number of columns (number of strips -1)

what is n in the trapezium rule?

n = number of columns (number of strips -1)

what values can we not integrate?

• sin²x

• cos²x

• cot²x

• sinxcosx

how do we integrate values we can’t integrate?

rewrite them using trig identities

how do you integrate sin²x ?

here

∫ sin²x dx

∫ sin²x dx = ½ x - ¼ x cosx + c

how do you integrate cos²x ?

here

∫ cos²x dx

here

how do you integrate cot²x ?

here

∫ cot²x dx

here

how do you integrate sinxcosx ?

here

∫ sinxcosx dx

here

INTEGRATION BY PARTS

∫ ab dx

∫ uv’ = uv - ∫ vu’

PRODUCT RULE

y = uv

dy / dx = vu’ + uv’

QUOTIENT RULE

y = u/v

dy / dx = (vu’ - uv’) / v²

∫ tanx dx

∫ tanx dx = ln|secx| + c

∫ - tanx dx

∫ - tanx dx = - ln|secx| + c

∫ secx dx

∫ secx dx = ln|secx + tanx| + c

∫ - secx dx

∫ - secx dx = - ln|secx + tanx| + c

∫ cotx dx

∫ cotx dx = ln|sinx| + c

∫ - cotx dx

∫ - cotx dx = - ln|sinx| + c

∫ cosecx dx

∫ cosecx dx = - ln|cosecx + cotx| + c

∫ - cosecx dx

∫ - cosecx dx = ln|cosecx + cotx| + c

how do you make a trapezium rule estimate more accurate?

• using more strips

• using more trapezia