calculus Flashcards
improper integral
the expression to be integrated is undefined within the limits of integration or at one or both of the limits
examples of improper integrals
- a limit is infinity
- 1/x is undefined at x=0
solving improper integrals with limits of infinity
- replace infinity with a variable, a
- evaluate the integral using a and the other limit
- see what happens when a -> ∞
- if the expression with a tends to 0 the value of the integral is the expression without a else the value can’t be found
solving improper integrals when the integral is undefined at a particular value
- split the integral at the undefined point
- replace the undefined limit with a a variable, a
- evaluate the integral(s) with a
- see what happens when a-> to the undefined value
derivative of arcsin x
1/ √(1-x^2)
derivative of arccos x
- 1/ √(1-x^2)
derivative of arctan x
1/ (1+x^2)
integral of 1/ √(a^2-x^2)
arcsin x/a + c
integral of 1/ (a^2+x^2)
1/a arctan x/a + c
volume of a solid formed by rotation about the x-axis
V = ∫ πy^2 dx
volume of a solid formed by rotation about the y-axis
V = ∫ πx^2 dy
mean value of a function
1/b-a ∫ f(x) dx
where b and a are the limits
integral of cosecxcotx
-cosecx + c
integral of secxtanx
secx + c
integral of cosec^2x
-cotx
derivative of tanx
sec^2x
when to substitue trigonometric functions
when there are fractional powers of (a+x^2)
volumes of revolution with two lines
square before you subtract
derivative of sinh x
cosh x
derivative of cosh x
sinh x
derivative of sin x
cos x
derivative of cos x
- sin x
derivative of arcsinh x
1/ √(1+x^2)
derivative of arccosh x
1/√(x^2 -1)
derivative of arctanh x
1/ (1-x^2)
integral of a 1/ √(a^2+x^2)
arcsinh(x/a) + c
integral of 1/√(x^2 -a^2)
arccosh(x/a) + c
integrating polar co-ordinates
1/2r^2dθ
