Integrate ∫ 1/(x^2√(9 - x^2)) dx Step by Step

Step by step

The question

Question: find the integral of 1/(x²√(9 − x²)).

∫ 1x²√9 − x² dx =

1 Choose the substitution

We have the root of 9 minus x².

Use x = 3 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 = 3 sin θ.

θ is an angle.

x = 3 sin θ

Its derivative: dx = 3 cos θ dθ.

dx = 3 cos θ dθ

Careful!

dx does not just become dθ.

dx is 3 cos θ dθ.

Careful: not dx = dθ ✗

2 Simplify the root

Put x = 3 sin θ in: x² is 9 sin² θ.

√9 − x² = √9 − 9sin² θ

Take out 9: its root is 3.

= 3√1 − sin² θ

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

= 3 cos θ

1 − sin² θ = cos² θ

3 Write it with θ

x² is 9 sin² θ, and the root is 3 cos θ.

3 cos θ cancels top and bottom.

= ∫ 3 cos θ9 sin² θ × 3 cos θ dθ

= ∫ 19 sin² θ dθ

1 over sin² θ is csc² θ.

The 1/9 waits in front.

= 19∫ csc² θ dθ

4 Integrate

The derivative of cot θ is minus csc² θ.

So the integral of csc² θ is minus cot θ.

= −19cot θ + C

5 Back to x

Draw a right triangle for sin θ = x/3: opposite x, hypotenuse 3.

Pythagoras gives the third side: the root of 9 − x².

cot θ is adjacent over opposite.

cot θ = √9 − x²x

So the answer is minus the root of 9 − x², over 9x, + C.

= −√9 − x²9x + C

6 Check

Let's check: the slope of our answer F near x = 1, with a calculator: about 0.3536.

F(1.001) − F(0.999)0.002 ≈ 0.3536

The question at x = 1 gives 0.3536.

The same.

Right!

the question at x = 1: 1x²√9 − x² ≈ 0.3536

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