1
Q
Integrate ln u
A
u ln u - u
2
Q
Integrate 1 / ax+b
A
1/a ln(ax+b)
3
Q
Integrate cosh x
A
sinhx
4
Q
Integrate sinh x
A
coshx
5
Q
Integrate tanh x
A
ln|coshx|
6
Q
Integrate tan u
A
-ln|cos u|
7
Q
Integrate sec u
A
ln|sec u + tan u|
8
Q
Integrate sec x tan x
A
sec x
9
Q
Integrate csc x cot x
A
-csc x
10
Q
Integrate sec² x
A
tan x
11
Q
Integrate csc² x
A
-cot x
12
Q
Integrate sech² x
A
tanh x
13
Q
Integrate 1/ a²+x²
A
1/a tan-1(x/a)
14
Q
Integrate 1 / a²-x²
A
1/a tanh-1 x/a
15
Q
Integrate 1 / √x²-a²
A
cosh-1 x/a
16
Q
Integrate 1 / √x²+a²
A
sinh-1 x/a
17
Q
Integrate 1 / √a²-x²
A
sin-1 x/a or cos-1 x/a
18
Q
Integrate sin (A+ ½π)
A
= int cos A -> sin A
19
Q
How to solve trig functions at higher powers (find some examples)
A
20
Q
half angles
A
t = 2t/1-t^2
c = 1-t2 / 1+t2
s = 2t / 1+t2
tan x/2 = t
21
Q
integrate y / x²+y² dx =
A
tan-1 (x/y)
22
Q
sqrt(a²-x²)
A
asin, acos
23
Q
sqrt(x²-a²)
A
acosh
24
Q
sqrt(x²+a²)
A
asinh
25
a²+x²
atan
26
a²-x²
atanh
acoth
(a-x)(a+x)
27
Solve ∫e^ax cosx dx by complex by complex mixed trigo
Re[ int e^ax e^ix]
Re[e^(a+i)x / a+i]
e^ax / 1+a^2 (acos x + sin x)
28
I(q)= int from a(q) to b(q) f(x;q)dx
dI/dq = int df/dq dx + f(b;q) db/dq - f(a(q), q) da / dq
29
Solve int from 0 to π/2 sin^(2n)x dx
consider
sin^(2n-1) x d -cos x
30
Solve int from 0 to inf x^n e^-ax dx
n! / a^n+1
31
Prove stirling’s approximation
sum to n-1 ln k ≤ int to n ln x ≤ sum to n ln k
32
What is stirling’s approximation result
ln n! = n ln n - n
33
Prove schwarz’s inequality
prove by sum (a+b)^2 ≥ 0
34
How to simplify independent multiple integrals
can split
35
How to swap integrals if int from 1 to 2 then int from 0 to x
y ≤ x ≤ 2, 0 ≤ y ≤ 1
36
2D coordinates: x, y, z?
volume?
x = rcos
y = rsin
dA = r dr d0
37
Cylindrical polar: x, y, z?
volume?
x = rcos
y = rsin
dV = r dr d0 dz
38
Spherical polar: x, y, z?
volume?
x = r sin 0 cos ø
y = r sin 0 sin ø
z = r cos 0
dV = r^2 sin 0 dr d0 dø
39
Prove gaussian
sqrt(π/a)
