Integrate ∫ 1/(x^2√(4 - x^2)) dx Step by Step

Step by step

The question

Question: find the integral of 1/(x²√(4 − x²)).

∫ 1x²√4 − x² dx =

1 Choose the substitution

We have the root of 4 minus x².

Use x = 2 sin θ, because 1 − sin² θ is cos² θ: the minus goes away.

√a² − x²: x = a sin θ

a² + x²: x = a tan θ

So x = 2 sin θ.

θ is an angle.

x = 2 sin θ

Its derivative: dx = 2 cos θ dθ.

dx = 2 cos θ dθ

Careful!

dx does not just become dθ.

dx is 2 cos θ dθ.

Careful: not dx = dθ ✗

2 Simplify the root

Put x = 2 sin θ in: x² is 4 sin² θ.

√4 − x² = √4 − 4sin² θ

Take out 4: its root is 2.

= 2√1 − sin² θ

1 − sin² θ is cos² θ, and its root is cos θ.

= 2 cos θ

1 − sin² θ = cos² θ

3 Write it with θ

x² is 4 sin² θ, and the root is 2 cos θ.

2 cos θ cancels top and bottom.

= ∫ 2 cos θ4 sin² θ × 2 cos θ dθ

= ∫ 14 sin² θ dθ

1 over sin² θ is csc² θ.

The 1/4 waits in front.

= 14∫ csc² θ dθ

4 Integrate

The derivative of cot θ is minus csc² θ.

So the integral of csc² θ is minus cot θ.

= −14cot θ + C

5 Back to x

Draw a right triangle for sin θ = x/2: opposite x, hypotenuse 2.

Pythagoras gives the third side: the root of 4 − x².

cot θ is adjacent over opposite.

cot θ = √4 − x²x

So the answer is minus the root of 4 − x², over 4x, + C.

= −√4 − x²4x + C

6 Check

Let's check: the slope of our answer F near x = 1, with a calculator: about 0.5774.

F(1.001) − F(0.999)0.002 ≈ 0.5774

The question at x = 1 gives 0.5774.

The same.

Right!

the question at x = 1: 1x²√4 − x² ≈ 0.5774

Let's look back.

First: choose the substitution.

Then: simplify the root.

Then: write it with θ.

Then: integrate.

Then: back to x.

Last: check.

Answer