Trig Identities and Derivatives

Rewrite sin^2(x) using cos(x).

 ext{sin}^2(x) = 1 - ext{cos}^2(x)

Rewrite cos^2(x) using sin(x).

 ext{cos}^2(x) = 1 - ext{sin}^2(x)

Rewrite tan^2(x) using sec(x).

 ext{tan}^2(x) = ext{sec}^2(x) - 1

Rewrite cot^2(x) using csc(x).

 ext{cot}^2(x) = ext{csc}^2(x) - 1

What is sin(2x)?

 ext{sin}(2x) = 2* ext{sin}(x)* ext{cos}(x)

What is the first formula for cos(2x)?

 ext{cos}(2x) = ext{cos}^2(x) - ext{sin}^2(x)

What is cos(2x) in terms of cos(x)?

 ext{cos}(2x) = 2* ext{cos}^2(x) - 1

What is cos(2x) in terms of sin(x)?

 ext{cos}(2x) = 1 - 2* ext{sin}^2(x)

What is the power-reduction identity for sin^2(x)?

 ext{sin}^2(x) = rac{1 - ext{cos}(2x)}{2}

What is the power-reduction identity for cos^2(x)?

 ext{cos}^2(x) = rac{1 + ext{cos}(2x)}{2}

What is sin(A + B)?

 ext{sin}(A + B) = ext{sin}(A) ext{cos}(B) + ext{cos}(A) ext{sin}(B)

What is sin(A - B)?

 ext{sin}(A - B) = ext{sin}(A) ext{cos}(B) - ext{cos}(A) ext{sin}(B)

What is cos(A + B)?

 ext{cos}(A + B) = ext{cos}(A) ext{cos}(B) - ext{sin}(A) ext{sin}(B)

What is cos(A - B)?

 ext{cos}(A - B) = ext{cos}(A) ext{cos}(B) + ext{sin}(A) ext{sin}(B)

What is the antiderivative power rule?

 ext{Integral of } x^n ext{ dx } = rac{x^{n+1}}{n+1} + C, ext{ where } n ext{ is not } -1

What is the antiderivative of a constant k?

 ext{Integral of } k ext{ dx } = kx + C

What is the antiderivative of } rac{1}{x}?

 ext{Integral of } rac{1}{x} ext{ dx } = ext{ln}|x| + C

What is the antiderivative of e^x?

 ext{Integral of } e^x ext{ dx } = e^x + C

What is the antiderivative of a^x?

 ext{Integral of } a^x ext{ dx } = rac{a^x}{ ext{ln}(a)} + C

What is the antiderivative of sin(x)?

 ext{Integral of } ext{sin}(x) ext{ dx } = - ext{cos}(x) + C

What is the antiderivative of cos(x)?

 ext{Integral of } ext{cos}(x) ext{ dx } = ext{sin}(x) + C

What is the antiderivative of tan(x)?

 ext{Integral of } ext{tan}(x) ext{ dx } = ext{ln}| ext{sec}(x)| + C

What is the antiderivative of cot(x)?

 ext{Integral of } ext{cot}(x) ext{ dx } = ext{ln}| ext{sin}(x)| + C

What is the antiderivative of sec(x)?

 ext{Integral of } ext{sec}(x) ext{ dx } = ext{ln}| ext{sec}(x) + ext{tan}(x)| + C

What is the antiderivative of csc(x)?

 ext{Integral of } ext{csc}(x) ext{ dx } = ext{ln}| ext{csc}(x) - ext{cot}(x)| + C

What is the antiderivative of sec^2(x)?

 ext{Integral of } ext{sec}^2(x) ext{ dx } = ext{tan}(x) + C

What is the antiderivative of csc^2(x)?

 ext{Integral of } ext{csc}^2(x) ext{ dx } = - ext{cot}(x) + C

What is the antiderivative of sec(x)tan(x)?

 ext{Integral of } ext{sec}(x) ext{tan}(x) ext{ dx } = ext{sec}(x) + C

What is the antiderivative of csc(x)cot(x)?

 ext{Integral of } ext{csc}(x) ext{cot}(x) ext{ dx } = - ext{csc}(x) + C

What is the antiderivative of } rac{1}{1 + x^2}?

 ext{Integral of } rac{1}{1 + x^2} ext{ dx } = ext{arctan}(x) + C

What is the antiderivative of } rac{1}{ ext{sqrt}(1 - x^2)}?

 ext{Integral of } rac{1}{ ext{sqrt}(1 - x^2)} ext{ dx } = ext{arcsin}(x) + C

What is the antiderivative of } rac{1}{a^2 + x^2}?

 ext{Integral = } rac{1}{a} ext{arctan}igg(rac{x}{a}igg) + C, ext{ assuming } a > 0

What is the antiderivative of } rac{1}{ ext{sqrt}(a^2 - x^2)}?

 ext{Integral = } ext{arcsin}igg(rac{x}{a}igg) + C, ext{ assuming } a > 0

What is the reverse chain-rule logarithm pattern?

 ext{Integral of } rac{f'(x)}{f(x)} ext{ dx } = ext{ln}|f(x)| + C

What is the antiderivative of sin(ax + b)?

 ext{Integral = } -rac{1}{a} ext{cos}(ax + b) + C

What is the antiderivative of cos(ax + b)?

 ext{Integral = } rac{1}{a} ext{sin}(ax + b) + C

What is the antiderivative of sec^2(ax + b)?

 ext{Integral = } rac{1}{a} ext{tan}(ax + b) + C

What is the antiderivative of sec(ax + b)tan(ax + b)?

 ext{Integral = } rac{1}{a} ext{sec}(ax + b) + C

Why must indefinite integrals include + C?

Because the derivative of any constant is zero, so an antiderivative represents a family of functions.

What does a definite integral represent?

The net signed accumulation from the lower bound to the upper bound.

What is the Fundamental Theorem of Calculus evaluation rule?

If F'(x) = f(x), then the integral from a to b of f(x) ext{ dx } equals F(b) - F(a).

What is displacement from velocity v(t)?

Displacement = ext{integral from } a ext{ to } b ext{ of } v(t) ext{ dt }.

What is total distance from velocity v(t)?

Total distance = ext{integral from } a ext{ to } b ext{ of } |v(t)| ext{ dt }.