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

Step by step

The question

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

∫ 1x² + 9 dx =

1 Choose the substitution

We have 9 plus x².

Use x = 3 tan θ, because 1 + tan² θ is sec² θ: the plus goes away.

√a² − x²: x = a sin θ

a² + x²: x = a tan θ

So x = 3 tan θ.

θ is an angle.

x = 3 tan θ

Its derivative: dx = 3 sec² θ dθ.

dx = 3 sec² θ dθ

Careful!

dx does not just become dθ.

dx is 3 sec² θ dθ.

Careful: not dx = dθ ✗

2 Simplify the root

Put x = 3 tan θ in: x² is 9 tan² θ.

x² + 9 = 9tan² θ + 9

Take out 9.

= 9(1 + tan² θ)

1 + tan² θ is sec² θ.

= 9 sec² θ

1 + tan² θ = sec² θ

3 Write it with θ

dx on top, 9 sec² θ at the bottom.

sec² θ cancels.

= ∫ 3 sec² θ9 sec² θ dθ

3/9 is 1/3.

= ∫ 13 dθ

4 Integrate

The integral of a number is the number times θ.

= 13θ + C

5 Back to x

From x = 3 tan θ: tan θ is x/3.

tan θ = x3

So the answer is 1/3 times tan⁻¹(x/3) + C.

= 13tan⁻¹(x3) + C

6 Check

Let's check: the slope of our answer F near x = 1, with a calculator: about 0.1000.

F(1.001) − F(0.999)0.002 ≈ 0.1000

The question at x = 1 gives 0.1000.

The same.

Right!

the question at x = 1: 1x² + 9 ≈ 0.1000

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