calc 2 (21b) trig identities, integrals, & relevant formulas

∫ xⁿ dx

xⁿ⁺¹/(n+1) + C (n ≠ −1)

∫ sin(kx) dx

−(1/k) cos(kx) + C

∫ cos(kx) dx

(1/k) sin(kx) + C

∫ tan(kx) dx

(1/k) ln|sec(kx)| + C

∫ xⁿ dx

xⁿ⁺¹/(n+1) + C, n ≠ −1

∫ (1/x) dx

ln|x| + C

∫ sin(kx) dx

−(1/k) cos(kx) + C

∫ cos(kx) dx

(1/k) sin(kx) + C

∫ tan(kx) dx

(1/k) ln|sec(kx)| + C

∫ sec(kx) dx

(1/k) ln|sec(kx) + tan(kx)| + C

∫ sec(kx)tan(kx) dx

(1/k) sec(kx) + C

∫ sec²(kx) dx

(1/k) tan(kx) + C

∫ e^{kx} dx

(1/k) e^{kx} + C

∫ 1/(a² + x²) dx

(1/a) tan⁻¹(x/a) + C

∫ 1/√(a² − x²) dx

sin⁻¹(x/a) + C

∫ 1/(x√(x² − a²)) dx

(1/a) sec⁻¹(|x/a|) + C

Σ k

n(n+1)/2

Σ k²

n(n+1)(2n+1)/6

Σ k³

[n(n+1)/2]²

Average value of f

(1/(b−a)) ∫ₐᵇ f(x) dx

FTC Part I

d/dx ∫ₐˣ f(t) dt = f(x)

FTC Part II

∫ₐᵇ f(x) dx = F(b) − F(a)

u-substitution

∫ f(g(x))g′(x) dx = ∫ f(u) du

Even function integral

2∫₀ᵃ f(x) dx

Odd function integral

0

Area between curves (x)

∫ (top − bottom) dx

Area between curves (y)

∫ (right − left) dy

Disk method

π ∫ R(x)² dx

Washer method

π ∫ (R² − r²) dx

Shell method

2π ∫ (radius)(height) dx

Arc length (x)

∫ √(1 + (dy/dx)²) dx

Arc length (y)

∫ √(1 + (dx/dy)²) dy

Surface area about x-axis

2π ∫ f(x) √(1 + (f′(x))²) dx

Definition of ln(x)

∫₁ˣ 1/t dt

Derivative of aˣ

aˣ ln(a)

Integration by parts

∫ u dv = uv − ∫ v du

sin²x

(1 − cos(2x))/2

cos²x

(1 + cos(2x))/2

sin(2x)

2 sin x cos x

cos(2x)

cos²x − sin²x

1 + tan²x

sec²x

1 + cot²x

csc²x

Trig sub: √(a² − x²)

x = a sinθ

Trig sub: √(a² + x²)

x = a tanθ

Trig sub: √(x² − a²)

x = a secθ

Trapezoidal Rule

(Δx/2)[f(x₀)+2Σf(xᵢ)+f(xₙ)]

Simpson’s Rule

(Δx/3)[f(x₀)+4Σ f(odd)+2Σ f(even)+f(xₙ)]

Limit Comparison Test

lim (f(x)/g(x)) = L > 0 → same behavior

dy/dx for parametric

(dy/dt)/(dx/dt)

Arc length (parametric)

∫ √[(dx/dt)² + (dy/dt)²] dt

Area (parametric)

∫ y(t)x′(t) dt

Polar to Cartesian

x = r cosθ, y = r sinθ, tanθ=y/x