New Calc

∫ sin(x) dx

-cos(x) + C

∫ cos(x) dx

sin(x) + C

∫ sec²(x) dx

tan(x) + C

∫ csc²(x) dx

-cot(x) + C

∫ sec(x)tan(x) dx

sec(x) + C

∫ csc(x)cot(x) dx

-csc(x) + C

∫ tan(x) dx

ln|sec(x)| + C

∫ cot(x) dx

ln|sin(x)| + C

∫ sec(x) dx

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

∫ csc(x) dx

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

∫ tan³(x) dx

(1/2)tan²(x) - ln|sec(x)| + C

∫ sec³(x) dx

(1/2)[sec(x)tan(x) + ln|sec(x) + tan(x)|] + C

sin²(x) + cos²(x)

1

1 + tan²(x)

sec²(x)

1 + cot²(x)

csc²(x)

sin²(x)

1 - cos²(x)

cos²(x)

1 - sin²(x)

tan²(x)

sec²(x) - 1

cot²(x)

csc²(x) - 1

sin²(x) half-angle

½ (1 - cos(2x))

cos²(x) half-angle

½ (1 + cos(2x))

sin(x)cos(x)

(1/2)sin(2x)

sin^m(x)cos^n(x): m odd

Save one sin(x), convert sin²(x) to 1 - cos²(x), u = cos(x)

sin^m(x)cos^n(x): n odd

Save one cos(x), convert cos²(x) to 1 - sin²(x), u = sin(x)

sin^m(x)cos^n(x): m and n both even

Use half-angle identities

tan^m(x)sec^n(x): n even

Save sec²(x), u = tan(x)

tan^m(x)sec^n(x): m odd

Save sec(x)tan(x), u = sec(x)

a² - x² trig substitution

x = a sin(θ)

a² + x² trig substitution

x = a tan(θ)

x² - a² trig substitution

x = a sec(θ)

x = a sin(θ): dx

a cos(θ)dθ

x = a tan(θ): dx

a sec²(θ)dθ

x = a sec(θ): dx

a sec(θ)tan(θ)dθ

sqrt(a² - x²) after trig substitution

a cos(θ)

sqrt(a² + x²) after trig substitution

a sec(θ)

sqrt(x² - a²) after trig substitution

a tan(θ)

x = a sin(θ) triangle

opposite = x, hypotenuse = a, adjacent = sqrt(a² - x²)

x = a tan(θ) triangle

opposite = x, adjacent = a, hypotenuse = sqrt(x² + a²)

x = a sec(θ) triangle

hypotenuse = x, adjacent = a, opposite = sqrt(x² - a²)

Distinct linear factors (x-a)(x-b)

A/(x-a) + B/(x-b)

Repeated linear factor (x-a)²

A/(x-a) + B/(x-a)²

Repeated linear factor (x-a)³

A/(x-a) + B/(x-a)² + C/(x-a)³

Irreducible quadratic factor x²+a²

(Ax+B)/(x²+a²)

Repeated irreducible quadratic (x²+a²)²

(Ax+B)/(x²+a²) + (Cx+D)/(x²+a²)²

Partial fractions degree rule

Degree of numerator must be less than degree of denominator

Improper integral p-test: ∫₁^∞ 1/x^p dx

p > 1 converges, p ≤ 1 diverges

Improper integral p-test: ∫₀¹ 1/x^p dx

p < 1 converges, p ≥ 1 diverges

Arc length with dx

L = ∫ sqrt(1 + (dy/dx)²) dx

Arc length with dy

L = ∫ sqrt(1 + (dx/dy)²) dy

Arc length element ds using dx

ds = sqrt(1 + (dy/dx)²) dx

Arc length element ds using dy

ds = sqrt(1 + (dx/dy)²) dy

Surface area master formula

S = 2π∫r ds

Surface area around x-axis using dx

S = 2π∫ y sqrt(1 + (dy/dx)²) dx

Surface area around y-axis using dx

S = 2π∫ x sqrt(1 + (dy/dx)²) dx

Surface area around x-axis using dy

S = 2π∫ y sqrt(1 + (dx/dy)²) dy

Surface area around y-axis using dy

S = 2π∫ x sqrt(1 + (dx/dy)²) dy

Rotation around x-axis: radius

r = |y|

Rotation around y-axis: radius

r = |x