Radical

Substitution

Resulting identity

\sqrt{a^2-x^2}

x=a\sin\theta

\sqrt{a^2-x^2}=a\cos\theta

\sqrt{a^2+x^2}

x=a\tan\theta

\sqrt{a^2+x^2}=a\sec\theta

\sqrt{x^2-a^2}

x=a\sec\theta

\sqrt{x^2-a^2}=a\tan\theta

Section 7.1

Substitution table

We assume $a>0$ and choose an interval of $\theta$ compatible with the signs.

Explore the idea

Trigonometric substitution removes the radical with Pythagorean identities. The reference triangle converts back to x.

Pick a radical and move θ: the sides and the simplification update together.

Substitution: x = a sen(θ)

dx = a cos(θ) dθ

√radical = √(a² − x²) = a cos(θ)

Restriction: θ ∈ [0, π/2], x ∈ [0, a]

a = 2 · θ = 0.7 rad (40.1°) · x = 1.2884

√(a² − a² sen²θ) = a √(1 − sen²θ) = a cos(θ)

a = 22

θ = 0.70.7

Show angle θShow algebraic simplification