Integrate ∫ 1/(x^2 + 4)^(3/2) dx Step by Step

Step by step

The question

Question: find the integral of 1/(x² + 4)^(3/2).

∫ 1(x² + 4)^(3/2) 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

To the power three halves: the root, then cubed.

The root of 4 sec² θ is 2 sec θ; cubed, 8 sec³ θ.

(x² + 4)^(3/2) = (4 sec² θ)^(3/2) = 8 sec³ θ

dx on top, 8 sec³ θ at the bottom.

= ∫ 2 sec² θ8 sec³ θ dθ

Cancel sec² θ and 2.

= 14∫ 1sec θ dθ

1 over sec θ is cos θ.

= 14∫ cos θ dθ

4 Integrate

The integral of cos θ is sin θ.

= 14sin θ + C

5 Back to x

Draw a right triangle for tan θ = x/2: opposite x, adjacent 2.

Pythagoras gives the hypotenuse: the root of x² + 4.

sin θ is opposite over hypotenuse.

sin θ = x√x² + 4

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

= x4√x² + 4 + C

6 Check

Let's check: the slope of our answer F near x = 1, with a calculator: about 0.0894.

F(1.001) − F(0.999)0.002 ≈ 0.0894

The question at x = 1 gives 0.0894.

The same.

Right!

the question at x = 1: 1(x² + 4)^(3/2) ≈ 0.0894

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