AP Calc-integrals

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42 Terms

1
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∫du
u + C
2
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∫edu
e^u + C
3
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∫cos(u)du
sin(u) + C
4
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∫cot(u)du
lnIsin(u)I + C
5
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∫csc(u)du
-lnIcsc(u) + cot(u)I + C
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∫csc²(u)du
-cot(u) + C
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∫csc(u)cot(u)du
-csc(u) + C
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∫du/(a²+u²)
(1/a)arctan(u/a) + C
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∫[f(u) + g(u)]du
∫f(u)du + ∫g(u)du
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∫[f(u) - g(u)]du
∫f(u)du - ∫g(u)du
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∫(a^u)du
(1/ln(a))a^u +C
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∫sin(u)du
-cos(u) + C
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∫tan(u)du
-lnIcos(u)I + C
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∫sec(u)du
lnIsec(u) + tan(u)I + C
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∫sec²(u)du
tan(u) + C
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∫sec(u)tan(u)du
sec(u) + C
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∫du/√(a²-u²)
arcsin(u/a) +C
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∫du/[(u)√(u²−a²)
(1/a)arcsec(IuI/a) + C
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net change formula
∫f'(x)=f(b) - f(a)
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when do you use net change formula?
when the question asks for a rate of change
21
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∆x=?
(b-a)/n
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xi=
a + i∆x
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area under the curve=
∆x(∑heights)
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trapezoidal rule
(b-a)/(2n)[f(a) + 2f(n-1) + f(b)]
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how do you do a midpoint sum
you take the ∆x of the points and multiply it by the middle number
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∑c=
cn
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n(n+2)/2
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∑i²=
n(n+1)(2n+1)/6
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∑i³=
n²(n+1)²/4
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∑(ai + bi)=
∑ai + ∑bi
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∫f(x)dx=
limn→∞∑f(xi)∆x
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(d/dx)g(x)=∫(from x to 0)√1+t² dt(d/dx)=
g'(x)=√1+x² (1) - √1+0² (0)
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∫x^n=
(x^n+1)/(n+1)
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∫(from a to a)f(x)dx=
0
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∫(from b to a)f(x)dx=
-∫(from a to b)f(x)dx
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f(avg)=
1/(b-a)∫(from a to b)f(x)dx
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steps for u-sub
1. choose u
2. find du
3. rewrite integral in terms of u
4. evaluate integral
5. replace u with function
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if f is even, f(-x)=f(x), then ∫(from -a to a)f(x)dx=
2∫(from 0 to a)f(x)dx
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if f is odd, f(-x)=-f(x), then ∫(from -a to a)f(x)dx=
0
40
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when n/d, n is 2 less, use...
inverse trig
41
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when n/d, n=d, use...
u-sub
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when n/d, n≥d, use...
long division