Integrals Flashcards
integral of cos(x)dx =
sin(x) + c
integral of sin(x)dx =
-cos(x) +c
∫a dx =
ax + C
∫x dx
x2/2 + C
∫x2 dx
x3/3 + C
∫(1/x) dx
ln|x| + C
∫ex dx
ex + C
∫ax dx
ax/ln(a) + C
∫ln(x) dx
x ln(x) − x + C
∫cos(x) dx
sin(x) + C
∫sin(x) dx
-cos(x) + C
∫sec2(x) dx
tan(x) + C
∫cf(x) dx
c∫f(x) dx
∫x^n dx
(x^n+1)/(n+1) + C
∫(f + g) dx
∫f dx + ∫g dx
∫(f - g) dx
∫f dx - ∫g dx
When doing integration by parts, a good rule of thumb is to …
choose a u that gets simpler when you differentiate it and a v that doesn’t get any more complicated when you integrate it.
A helpful rule of thumb when integrating by parts is ______. Choose u based on which of these comes first:
I LATE
I: Inverse trigonometric functions such as sin^-1(x), cos^-1(x), tan^-1(x)
L: Logarithmic functions such as ln(x), log(x)
A: Algebraic functions such as x2, x3
T: Trigonometric functions such as sin(x), cos(x), tan (x)
E: Exponential functions such as ex, 3x
sin^2(x) + cos^2(x) =
1
Partial Fraction decomposition only works when _______________, and if it’s not in this form, we need to _______________ first.
the degree of the top is less than the bottom.
do polynomial long division first.
In partial fraction decomposition, we reduce until we have complex numbers.
FALSE
we reduce until we get an irreducible form (a fraction that if further reduced breaks into complex numbers)
In partial fraction decomposition, When you have a quadratic factor you need to include this partial fraction:
(B1x + C1) / (Your Quadratic)
In partial fraction decomposition, sometimes you may get a factor with an exponent, like (x−2)^3. When this happens, you …
The same thing can also happen to quadratics:
1/(x2+2x+3)^2 has the partial fractions …
need a partial fraction for each exponent from 1 up.
A1/(x−2) + A2/(x−2)^2 + A3/(x−2)^3
(B1x + C1)/(x2+2x+3) + (B2x + C2)/(x2+2x+3)^2
In partial fraction decomposition, even after using the roots (zeros) of the bottom you can end up with unknown constants.
So the next thing to do is:
Gather all powers of x together and then solve it as a system of linear equations.
