Exam (elaborations)
Calculus II - the review set Questions and and answers (1)
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Calculus II - the review set Questions and and answers (1)
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Calculus II - the review setd/dx sin x - ✅✅-cos x
d/dx cos x - ✅✅--sin x
d/dx tan x - ✅✅-sec² x
d/dx cot x - ✅✅--csc² x
d/dx sec x - ✅✅-sec x tan x
d/dx csc x - ✅✅--csc x cot x
∫cos x dx - ✅✅-sin x + C
∫sin x dx - ✅✅--cos x + C
∫sec²x dx - ✅✅-tan x + C
∫csc² x dx - ✅✅--cot x + C
∫(sec x tan x) dx - ✅✅-sec x + C
∫(csc x cotx) dx - ✅✅--csc x + C
∫tan x dx - ✅✅-ln|sec x| + C
(think of it from ln|1/cosx| + 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
∫ dx - ✅✅-x + C
for definite integrals:
∫(from a to b) f(x)dx - ✅✅-F(b) - F(a)
(top - bottom)
, d/dx ln(x) - ✅✅-1/x
∫ (1/x) dx - ✅✅-ln|x| + C
d/dx (a^x) - ✅✅-(a^x) . (ln(a))
∫ (a^x) dx - ✅✅-((a^x)/(ln(a))) + C
when do we use integration by parts? - ✅✅ --when it is a composition function
(meaning two different type of functions are being multiplied together)
-they're being multiplied
multiplied + two different type of functions
integration by parts formula? - ✅✅-∫u dv = u.v - ∫v du
(sudeve = uvulerek - suvadustu)
comes from taking the antiderivative of the product formula: ∫d/dx f.g = ∫ f'.g + ∫ f.g'
✅✅
What is the keyword and the explanation of it, to find the order of u and dv while
doing integration by parts? - -LIATE
Logs
Inverse trig
Algebraic
Trig
Exponents
the first one in order is → u
the second one in order is → dv
[CAUTION: does NOT always work, you have to check whether it is the right choice
or not!!! sometimes you might have to choose the opposite of what the keyword
says]
for example:
∫ x.sinx dx
u = x (Algebraic - 1st)
dv = sinx dx (Trig - 2nd)
[we choose u and dv because of the formula and the fact that formula starts with
those two]
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