Integrate ∫ 1/(x^2 + 25) dx Step by Step

Step by step

The question

Question: find the integral of 1/(x² + 25).

∫ 1x² + 25 dx =

1 Choose the substitution

We have 25 plus x².

Use x = 5 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 = 5 tan θ.

θ is an angle.

x = 5 tan θ

Its derivative: dx = 5 sec² θ dθ.

dx = 5 sec² θ dθ

Careful!

dx does not just become dθ.

dx is 5 sec² θ dθ.

Careful: not dx = dθ ✗

2 Simplify the root

Put x = 5 tan θ in: x² is 25 tan² θ.

x² + 25 = 25tan² θ + 25

Take out 25.

= 25(1 + tan² θ)

1 + tan² θ is sec² θ.

= 25 sec² θ

1 + tan² θ = sec² θ

3 Write it with θ

dx on top, 25 sec² θ at the bottom.

sec² θ cancels.

= ∫ 5 sec² θ25 sec² θ dθ

5/25 is 1/5.

= ∫ 15 dθ

4 Integrate

The integral of a number is the number times θ.

= 15θ + C

5 Back to x

From x = 5 tan θ: tan θ is x/5.

tan θ = x5

So the answer is 1/5 times tan⁻¹(x/5) + C.

= 15tan⁻¹(x5) + C

6 Check

Let's check: the slope of our answer F near x = 1, with a calculator: about 0.0385.

F(1.001) − F(0.999)0.002 ≈ 0.0385

The question at x = 1 gives 0.0385.

The same.

Right!

the question at x = 1: 1x² + 25 ≈ 0.0385

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