Question

Find each integral. $\int \frac{d x}{x^{2}+4 x+8}$

   Find each integral.
$\int \frac{d x}{x^{2}+4 x+8}$

Calculus, Early Transcendentals

Calculus, Early Transcendentals

Michael Sullivan,… 1st Edition

Chapter 7, Problem 3

Instant Answer

Solved by Expert Ma. Theresa Alin

07/30/2021

Step 1

We can complete the square in the denominator to get $\displaystyle x^{2}+4x+4+4$, which simplifies to $\displaystyle ( x+2)^{2}+4$. So, the integral becomes \[ \int \frac{d x}{(x+2)^{2}+2^{2}} \] Show more…

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Find each integral. $\int \frac{d x}{x^{2}+4 x+8}$

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Key Concepts

Completing the Square

This algebraic technique involves rewriting a quadratic expression as a perfect square plus a constant. It is essential for transforming integrals of rational functions into a form where standard integration formulas, such as those involving inverse trigonometric functions, can be applied.

Standard Arctangent Integration Formula

The integration formula ? dx/(x² + a²) = (1/a) arctan(x/a) + C is frequently used after adjusting the integrand into the appropriate form. It is crucial for solving integrals that result from algebraic manipulations like completing the square.

Substitution Method

The substitution method simplifies an integral by changing variables, making the integrand easier to integrate. In the context of this problem, a substitution is often used after completing the square to center the variable, preparing the integral for direct application of the arctan formula.

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