Trig Sub

√(a² - x²)

Substitute: x = a sin θ

Differential: dx = a cos θ dθ

Limits: -π/2 ≤ θ ≤ π/2

Back substitute: θ = arcsin(x / a)

√(a² + x²)

Substitute: x = a tan θ

Differential: dx = a sec² θ dθ

Limits: -π/2 < θ < π/2

Back substitute: θ = arctan(x / a)

√(x² - a²)

Substitute: x = a sec θ

Differential: dx = a sec θ tan θ dθ

Limits: 0 ≤ θ < π/2 or π ≤ θ < 3π/2

Back substitute: θ = arcsec(x / a)

or

Substitute: x = a cosh u

Differential: dx = a sinh u du

Limits: u ≥ 0

Back substitute: u = arccosh(x / a)

√(a² - (bx)²)

Substitute: x = (a/b) sin θ

Differential: dx = (a/b) cos θ dθ

Limits: -π/2 ≤ θ ≤ π/2

Back substitute: θ = arcsin(bx / a)

√(a² + (bx)²)

Substitute: x = (a/b) tan θ

Differential: dx = (a/b) sec² θ dθ

Limits: -π/2 < θ < π/2

Back substitute: θ = arctan(bx / a)

√((bx)² - a²)

Substitute: x = (a/b) sec θ

Differential: dx = (a/b) sec θ tan θ dθ

Limits: 0 ≤ θ < π/2 or π ≤ θ < 3π/2

Back substitute: θ = arcsec(bx / a)

√(x² + a²)

Substitute: x = a sinh u

Differential: dx = a cosh u du

Limits: -∞ < u < ∞

Back substitute: u = arcsinh(x / a)