Integrate ∫ 1/(x^2√(25 - x^2)) dx Step by Step

Step by step

The question

Question: find the integral of 1/(x²√(25 − x²)).

∫ 1x²√25 − x² dx =

1 Choose the substitution

We have the root of 25 minus x².

Use x = 5 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 = 5 sin θ.

θ is an angle.

x = 5 sin θ

Its derivative: dx = 5 cos θ dθ.

dx = 5 cos θ dθ

Careful!

dx does not just become dθ.

dx is 5 cos θ dθ.

Careful: not dx = dθ ✗

2 Simplify the root

Put x = 5 sin θ in: x² is 25 sin² θ.

√25 − x² = √25 − 25sin² θ

Take out 25: its root is 5.

= 5√1 − sin² θ

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

= 5 cos θ

1 − sin² θ = cos² θ

3 Write it with θ

x² is 25 sin² θ, and the root is 5 cos θ.

5 cos θ cancels top and bottom.

= ∫ 5 cos θ25 sin² θ × 5 cos θ dθ

= ∫ 125 sin² θ dθ

1 over sin² θ is csc² θ.

The 1/25 waits in front.

= 125∫ csc² θ dθ

4 Integrate

The derivative of cot θ is minus csc² θ.

So the integral of csc² θ is minus cot θ.

= −125cot θ + C

5 Back to x

Draw a right triangle for sin θ = x/5: opposite x, hypotenuse 5.

Pythagoras gives the third side: the root of 25 − x².

cot θ is adjacent over opposite.

cot θ = √25 − x²x

So the answer is minus the root of 25 − x², over 25x, + C.

= −√25 − x²25x + C

6 Check

Let's check: the slope of our answer F near x = 1, with a calculator: about 0.2041.

F(1.001) − F(0.999)0.002 ≈ 0.2041

The question at x = 1 gives 0.2041.

The same.

Right!

the question at x = 1: 1x²√25 − x² ≈ 0.2041

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