Trig Subsitition

When do we use trig substitution?

\sqrt{a^2-x^2},\sqrt{a^2+x^2},\sqrt{x^2-a^2}

\sqrt{a^2-x^2}

x = a sinθ ; dx = a cosθ dθ

What does √(a² − x²) become after substitution?

a cosθ

\sqrt{a^2+x^2}

x = a tanθ ; dx = a sec²θ dθ

What does √(a² + x²) become after substitution?

a secθ

\sqrt{x^2-a^2}

x = a secθ ; dx = a secθ tanθ dθ

What does √(x² − a²) become after substitution?

a tanθ

Key trig identity used in trig substitution

sin²θ + cos²θ = 1

Why trig substitution works

It converts square roots using trig identities like √(1 − sin²θ) = cosθ and √(sec²θ − 1) = tanθ

Final step after trig substitution

Convert θ back to x using a right triangle

How to remember substitutions

√(a² − x²) → sin ; √(a² + x²) → tan ; √(x² − a²) → sec