Integrate ∫ 1/(x^2 + 9)^(3/2) dx Step by Step

Step by step

The question

Question: find the integral of 1/(x² + 9)^(3/2).

∫ 1(x² + 9)^(3/2) 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

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

The root of 9 sec² θ is 3 sec θ; cubed, 27 sec³ θ.

(x² + 9)^(3/2) = (9 sec² θ)^(3/2) = 27 sec³ θ

dx on top, 27 sec³ θ at the bottom.

= ∫ 3 sec² θ27 sec³ θ dθ

Cancel sec² θ and 3.

= 19∫ 1sec θ dθ

1 over sec θ is cos θ.

= 19∫ cos θ dθ

4 Integrate

The integral of cos θ is sin θ.

= 19sin θ + C

5 Back to x

Draw a right triangle for tan θ = x/3: opposite x, adjacent 3.

Pythagoras gives the hypotenuse: the root of x² + 9.

sin θ is opposite over hypotenuse.

sin θ = x√x² + 9

So the answer is x, over 9 times the root of x² + 9, + C.

= x9√x² + 9 + C

6 Check

Let's check: the slope of our answer F near x = 1, with a calculator: about 0.0316.

F(1.001) − F(0.999)0.002 ≈ 0.0316

The question at x = 1 gives 0.0316.

The same.

Right!

the question at x = 1: 1(x² + 9)^(3/2) ≈ 0.0316

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