trig integration identities

sine²(x) pythagorean

sin²(x) = 1 - cos²(x)

cos²(x) pythagorean

cos²(x) = 1 - sin²(x)

tan²(x) pythagorean

tan²(x) = sec²(x) - 1

sec²(x) pythagorean

sec²(x) = 1 + tan²(x)

cot²(x) pythagorean

cot²(x) = csc²(x) - 1

csc²(x) pythagorean

csc²(x) = 1 + cot²(x)

sin(2x)

sin(2x) = 2sin(x)cos(x)

cos(2x)

cos(2x) = cos²(x) - sin²(x)

sin(x ± y)

sin(x ± y) = sin(x)cos(y) ± sin(y)cos(x)

cos(x ± y)

cos(x)cos(y) ∓ sin(x)sin(y)

cos(x + y) = cos(x)cos(y) - sin(x)sin(y)

sin²(x) power reducing

sin²(x) = ½ (1 - cos(2x))

cos²(x) power reducing

cos²(x) = ½ (1 + cos(2x))

sin(mx)sin(nx)

sin(mx)sin(nx) = ½ [cos((m - n)x) - cos((m + n)x)]

sin(mx)cos(nx)

sin(mx)cos(nx) = ½ [sin((m - n)x) + sin((m + n)x)]

cos(mx)cos(nx)

cos(mx)cos(nx) = ½ [cos((m - n)x) + cos((m + n)x)]

√a² - x² trig substitution

x = asin(θ)

√a² + x² trig substitution

x = atan(θ)

√x² - a² trig substitution

x = asec(θ)

range for inverse sine

[-π/2, π/2]

range for inverse tangent

[-π/2, π/2]

range for inverse secant

[0, π/2) U (π/2, π]

√a² - x² simplifies to __ after trig substitution

a|cos(θ)|

√a² + x² simplifies to __ after trig substitution

asec(θ)

√x² - a² simplifies to __ after trig substitution

a|tan(θ)|

sin(-x)

-sin(x)

cos(-x)

cos(x)

tan(-x)

-tan(x)

sin(x/2) half angle identity

sin(x/2) = ±√½ (1 - cos(θ))

± based on the quadrant x/2 is in

cos(x/2) half angle identity

cos(x/2) = ±√½ (1 + cos(θ))

± based on the quadrant x/2 is in

tan(x/2) half angle identity

tan(x/2) = 1 - cos(θ) / sin(θ)

± based on the quadrant x/2 is in

∫ sin(x) dx

-cos(x) + c

∫ cos(x) dx

sin(x) + c

derivative of sine

cosine

derivative of cosine

negative sine

∫ sec²(x) dx

tan(x) + c

∫ sec(x)tan(x) dx

sec(x) + c

∫ csc²(x) dx

-cot(x) + c

∫ csc(x)cot(x)

-csc(x) + c

∫ tan(x) dx

-ln|cos(x)| + c

∫ cot(x) dx

ln|sin(x)| + c

∫ sec(x) dx

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

∫ csc(x)

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

∫ du/√a² - u²

arcsin(u/a) + c

∫ du/a² + u²

1/a * arctan(u/a) + c

∫ du/|u|√u² - a²

1/a * arcsec(|u|/a) + c