Integrate x / (x^2 + 2x + 4) - Step-by-Step Solution with Substitution

Integration of x / (x^2 + 2x + 4) using Substitution

This guide demonstrates how to integrate the expression x / (x^2 + 2x + 4) using the substitution method.

1. Substitution:

Let's start by substituting:

• u = x^2 + 2x + 4

2. Differentiate:

Now, differentiate both sides with respect to x:

• du/dx = 2x + 2

Solving for dx, we get:

• dx = du / (2x + 2)

3. Substitute into the Integral:

Substitute the values of u and dx into the original integral:

• ∫ x / (x^2 + 2x + 4) dx

• = ∫ x / u * (du / (2x + 2))

• = 1/2 ∫ (1/u) du

4. Integrate:

Integrate the simplified expression:

• = 1/2 ln|u| + C

5. Substitute Back:

Finally, substitute back the original value of u:

• = 1/2 ln|x^2 + 2x + 4| + C

Therefore, the integral of x / (x^2 + 2x + 4) is equal to 1/2 ln|x^2 + 2x + 4| + C, where C is the constant of integration.