Integrating -2(x+4)/(x^2+2x+4) | Step-by-Step Solution

Integrating -2(x+4)/(x^2+2x+4) - Step-by-Step Solution

This guide provides a step-by-step solution for integrating the expression -2(x+4)/(x^2+2x+4). We'll use logarithmic integration to solve this problem.

1. Recognizing the Form

The integrand -2(x+4)/(x^2+2x+4) resembles the derivative of a logarithmic function. Notice that the numerator (x+4) is almost the derivative of the denominator (x^2+2x+4).

2. Applying Logarithmic Integration

We can rewrite the integral using logarithmic integration. Let's break it down:

• Factor out a constant: -2(x+4)/(x^2+2x+4) = -2 * (x+4)/(x^2+2x+4)
• Recognize the derivative: The derivative of (x^2+2x+4) is (2x+2), which is similar to the numerator (x+4).
• Adjust the numerator: We can rewrite (x+4) as (1/2)*(2x+2) + 3

Now, the integral becomes:

∫-2 * [(1/2)*(2x+2) + 3] / (x^2+2x+4) dx

• Split the integral: We can separate this into two integrals:

∫-2 * (1/2)*(2x+2) / (x^2+2x+4) dx + ∫-2 * 3 / (x^2+2x+4) dx

• Simplify:

∫-(2x+2) / (x^2+2x+4) dx - 6∫1 / (x^2+2x+4) dx

• Apply logarithmic integration: The first integral is a straightforward logarithmic integration:

-ln|x^2+2x+4| - 6∫1 / (x^2+2x+4) dx

3. Completing the Square

The second integral requires completing the square in the denominator:

• Factor out a constant:

-6∫1 / (x^2+2x+4) dx = -6∫1 / [(x^2+2x+1) + 3] dx

• Complete the square: (x^2+2x+1) is a perfect square trinomial, (x+1)^2.

-6∫1 / [(x+1)^2 + 3] dx

4. Using a Trigonometric Substitution

We can use a trigonometric substitution to solve the remaining integral:

• Let: x+1 = √3 * tan(θ)
• Then: dx = √3 * sec^2(θ) dθ

Substituting these values, we get:

-6∫(√3 * sec^2(θ)) / [3tan^2(θ) + 3] dθ = -2∫sec^2(θ) / [tan^2(θ) + 1] dθ

• Simplify: Using the trigonometric identity tan^2(θ) + 1 = sec^2(θ), we get:

-2∫dθ = -2θ

5. Reverse Substitution

Substitute back for θ using x+1 = √3 * tan(θ):

-2θ = -2 * arctan[(x+1)/√3]

6. Final Solution

Combining all the parts, the final solution for the integral is:

∫-2(x+4)/(x^2+2x+4) dx = -ln|x^2+2x+4| - 2 * arctan[(x+1)/√3] + C

where C is the constant of integration.