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

Step by step

The question

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

∫ 1x² + 4 dx =

1 Choose the substitution

We have 4 plus x².

Use x = 2 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 = 2 tan θ.

θ is an angle.

x = 2 tan θ

Its derivative: dx = 2 sec² θ dθ.

dx = 2 sec² θ dθ

Careful!

dx does not just become dθ.

dx is 2 sec² θ dθ.

Careful: not dx = dθ ✗

2 Simplify the root

Put x = 2 tan θ in: x² is 4 tan² θ.

x² + 4 = 4tan² θ + 4

Take out 4.

= 4(1 + tan² θ)

1 + tan² θ is sec² θ.

= 4 sec² θ

1 + tan² θ = sec² θ

3 Write it with θ

dx on top, 4 sec² θ at the bottom.

sec² θ cancels.

= ∫ 2 sec² θ4 sec² θ dθ

2/4 is 1/2.

= ∫ 12 dθ

4 Integrate

The integral of a number is the number times θ.

= 12θ + C

5 Back to x

From x = 2 tan θ: tan θ is x/2.

tan θ = x2

So the answer is 1/2 times tan⁻¹(x/2) + C.

= 12tan⁻¹(x2) + C

6 Check

Let's check: the slope of our answer F near x = 1, with a calculator: about 0.2000.

F(1.001) − F(0.999)0.002 ≈ 0.2000

The question at x = 1 gives 0.2000.

The same.

Right!

the question at x = 1: 1x² + 4 ≈ 0.2000

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