Studied by 7 people∫0dx =
C
∫[g(x) + h(x)]dx =
G(x) + H(x) + C
∫xⁿdx =
1/n+1 * x^(n+1) + C
∫e^xdx =
e^x + C
∫sin(x)dx =
-cos(x) + C
∫sec²(x)dx =
tan(x) + C
∫sec(x)tan(x)dx =
sec(x) + C
∫⅟√(1-x²)dx =
arcsin(x) + C
∫kdx =
kx + C
∫kf(x)dx =
kF(x) + C
∫1/x dx =
ln(abs(x)) + C
∫a^x dx =
a^x / lna + C
∫cos(x)dx =
sin(x) + C
∫csc²(x)dx =
-cot(x) + C
∫csc(x)cot(x)dx =
-csc(x) + C
∫⅟(1+x²) dx =
arctan(x) + C
∫tan(x)dx =
ln(abs(sec(x))) + C
∫cot(x)dx =
ln(abs(sin(x))) + C
∫sec(x)dx =
ln(abs(secx + tanx)) + C
∫csc(x)dx =
ln(abs(cscx - cotx)) + C
∫ln(x)dx =
xlnx - x + C
substitution for √a²-x²
x = a*sin(theta)
substitution for √a²+x²
x = a*tan(theta)
substitution for √x²-a²
x = a*sec(theta)
1-sin²x =
cos^2 (x)
1 + tan²x =
sec^2 (x)
sec²x - 1 =
tan^2 (x)
√a²+x² becomes
a*sec(theta)
√x²-a² becomes
a*tan(theta)
√a²-x² becomes
a*cos(theta)
Integration by parts
integral(udv) = u*v - integral(v*du)
LIATE
Logarithm, Inverse trig, Algebraic, Trigonometric, Exponential
sin(2x) =
2sin(x)cos(x)
sin²x (power reduction) =
(1 - cos(2x))/2
cos²x (power reduction) =
(1 + cos(2x))/2
sinAcosB =
1/2(sin(A+B) + sin(A-B))
sinAsinB =
1/2(cos(A-B) - cos(A+B))
cosAcosB = `
1/2(cos(A-B) + cos(A+B))
∫⅟(a²+x²) dx =
1/a * arctan(x/a) + C
average value of f on [a, b]
1/(b-a) * integral from a to b of [f(x)dx]
Arc length L of f(x) on [a, b]
L = ∫[sqrt(1 + (dy/dx)^2) dx] on [a,b]
Surface area in terms of x across the x-axis
SA = ∫[2πf(x) * sqrt(1 + (dy/dx)^2) dx] on [a,b]
Surface area in terms of x across the y-axis
SA = integral from a to b of [2 pi x * sqrt(1 + (dy/dx)^2) dx]
Surface area in terms of y across the x-axis
SA = integral from a to b of [2 pi y * sqrt(1 + (dx/dy)^2) dy]
Surface area in terms of y across the y-axis
SA = integral from a to b of [2 pi g(y) * sqrt(1 + (dx/dy)^2) dy]
Note
Flashcard