Calculus 2 Formula Sheet

These flashcards cover key derivatives, integrals, and concepts from Calculus 2, providing definitions and rules related to trigonometric functions, logarithmic integration, and volume calculation methods.

d/dx [sin(x)]

cos(x)

d/dx [cos(x)]

-sin(x)

d/dx [tan(x)]

sec^2(x)

d/dx [cot(x)]

-csc^2(x)

d/dx [sec(x)]

sec(x)*tan(x)

d/dx [csc(x)]

-csc(x)*cot(x)

Integral 1/x dx

ln|x| + C

Integral f'(x)/f(x) dx

ln|f(x)| + C

Integral ln(x) dx

x*ln(x) - x + C

Integral a^x dx

a^x / ln(a) + C (where a > 0 and a != 1)

Integral e^x dx

e^x + C

Integral 1/x^2 dx

-1/x + C

Integral sec(x) dx

ln|sec(x) + tan(x)| + C

Integral csc(x) dx

-ln|csc(x) + cot(x)| + C

Integral sec^2(x) dx

tan(x) + C

Integral csc^2(x) dx

-cot(x) + C

Integral sec(x)*tan(x) dx

sec(x) + C

Integral csc(x)*cot(x) dx

-csc(x) + C

Integral 1/sqrt(1 - x^2) dx

arcsin(x) + C

Integral 1/(x^2 + a^2) dx

(1/a) * arctan(x/a) + C

Integral 1/(x*sqrt(x^2 - a^2)) dx

(1/a) * arcsec(|x|/a) + C

d/dx [arcsin(u)]

u' / sqrt(1 - u^2)

d/dx [arccos(u)]

-u' / sqrt(1 - u^2)

d/dx [arctan(u)]

u' / (1 + u^2)

d/dx [arccot(u)]

-u' / (1 + u^2)

d/dx [arcsec(u)]

u' / (|u| * sqrt(u^2 - 1))

d/dx [arccsc(u)]

-u' / (|u| * sqrt(u^2 - 1))

Washer Method for Volume

V = pi * Integral from a to b of [(R(x))^2 - (r(x))^2] dx

Disk Method

Washer Method with r(x) = 0

Washer/Disk Radius Rule (x-axis, region below)

radius = axis - function

Washer/Disk Radius Rule (x-axis, region above)

radius = function - axis

Shell Method Volume

V = 2*pi * Integral from a to b of r(x)*h(x) dx (vertical axis)

Shell Radius Rule (Vertical, region left)

radius = axis - x

Arc Length Formula

L = Integral from a to b of sqrt(1 + (f'(x))^2) dx

Work (Variable Force)

W = Integral from a to b of F(x) dx

Hooke's Law

F = k * x

Newton's Law of Gravitation

F(x) = C / x^2