Trigonometric Substitution

If an integral has a² - x², especially √a² - x²,
then set x = a sin θ.

Differentiate both sides.

Then dx = a cos θ dθ.

Derivative of a Composite Function

Derivative of sin x

Before using x and dx,
write an expression for θ.

Start from x = a sin θ.
sin θ = x/a
So θ = arcsin x/a.

Arcsine: Value

x = a sin θ
dx = a cos θ dθ

Put these into the given integral.

Then (given) = ∫ [a cos θ]/√a² - a² sin² θ dθ.

Integral by Substitution: Indefinite Integral

a² - a² sin² θ = a²(1 - sin² θ)

Common Monomial Factor

1 - sin² θ = cos² θ

Pythagorean Identity

√a² cos² θ = a cos θ

Simplify a Radical

∫ [a cos θ]/[a cos θ] dθ
= ∫ 1 dθ
= ∫ dθ

∫ dθ = θ + C

Integral of a Polynomial

Change θ back to x.

θ = arcsin x/a

So θ + C = arcsin x/a + C.

So
arcsin x/a + C
is the answer.

If an integral has a² + x²,
then set x = a tan θ.

Differentiate both sides.

Then dx = a sec² θ dθ.

Derivative of tan x

Before using x and dx,
write an expression for θ.

Start from x = a tan θ.
tan θ = x/a
So θ = arctan x/a.

Arctangent: Value

a tan θ
a sec² θ

Put these into the given integral.

Then (given) = ∫ [a sec² θ]/[a² + a² tan² θ] dθ.

a² + a² tan² θ = a²(1 + tan² θ)

1 + tan² θ = sec² θ

Pythagorean Identity

∫ [a sec² θ]/[a² sec² θ] dθ = ∫ 1/a dθ

Take 1/a out from the integral.

[1/a]∫ dθ = [1/a]θ + C

Change θ back to x.

θ = arctan x/a

So [1/a]θ + C = [1/a]arctan x/a + C.

So
[1/a]arctan x/a + C
is the answer.