Integrate ∫ 1/(x^2 + 16)^(3/2) dx Step by Step

Step by step

The question

Question: find the integral of 1/(x² + 16)^(3/2).

∫ 1(x² + 16)^(3/2) dx =

1 Choose the substitution

We have 16 plus x².

Use x = 4 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 = 4 tan θ.

θ is an angle.

x = 4 tan θ

Its derivative: dx = 4 sec² θ dθ.

dx = 4 sec² θ dθ

Careful!

dx does not just become dθ.

dx is 4 sec² θ dθ.

Careful: not dx = dθ ✗

2 Simplify the root

Put x = 4 tan θ in: x² is 16 tan² θ.

x² + 16 = 16tan² θ + 16

Take out 16.

= 16(1 + tan² θ)

1 + tan² θ is sec² θ.

= 16 sec² θ

1 + tan² θ = sec² θ

3 Write it with θ

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

The root of 16 sec² θ is 4 sec θ; cubed, 64 sec³ θ.

(x² + 16)^(3/2) = (16 sec² θ)^(3/2) = 64 sec³ θ

dx on top, 64 sec³ θ at the bottom.

= ∫ 4 sec² θ64 sec³ θ dθ

Cancel sec² θ and 4.

= 116∫ 1sec θ dθ

1 over sec θ is cos θ.

= 116∫ cos θ dθ

4 Integrate

The integral of cos θ is sin θ.

= 116sin θ + C

5 Back to x

Draw a right triangle for tan θ = x/4: opposite x, adjacent 4.

Pythagoras gives the hypotenuse: the root of x² + 16.

sin θ is opposite over hypotenuse.

sin θ = x√x² + 16

So the answer is x, over 16 times the root of x² + 16, + C.

= x16√x² + 16 + C

6 Check

Let's check: the slope of our answer F near x = 1, with a calculator: about 0.0143.

F(1.001) − F(0.999)0.002 ≈ 0.0143

The question at x = 1 gives 0.0143.

The same.

Right!

the question at x = 1: 1(x² + 16)^(3/2) ≈ 0.0143

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