Evaluate the following integrals.
∫ x/(x² + 6x + 18) dx

Verified step by step guidance

1. Rewrite the denominator of the integrand, $x²\; +\; 6x\; +\; 18$, as a perfect square trinomial. Complete the square: $x²\; +\; 6x\; +\; 18\; =\; (x\; +\; 3)²\; +\; 9$.
2. Substitute $u\; =\; x\; +\; 3$, which implies $du\; =\; dx$. The integral becomes $\int \; (u\; -\; 3)/((u²\; +\; 9))\; du$.
3. Split the integrand into two separate terms: $\int \; u/(u²\; +\; 9)\; du\; -\; \int \; 3/(u²\; +\; 9)\; du$. This allows us to handle each term individually.
4. For the first term, $\int \; u/(u²\; +\; 9)\; du$, use the substitution $v\; =\; u²\; +\; 9$, which implies $dv\; =\; 2u\; du$. Rewrite the integral accordingly.
5. For the second term, $\int \; 3/(u²\; +\; 9)\; du$, recognize it as a standard integral of the form $\int \; 1/(u²\; +\; a²)\; du$, which evaluates to $(1/a)\; arctan(u/a)$. Apply this formula with $a\; =\; 3$.

Key Concepts

Integration

Integration is a fundamental concept in calculus that involves finding the integral of a function, which represents the area under the curve of that function. It is the reverse process of differentiation and can be used to calculate quantities such as areas, volumes, and accumulated values. Understanding the techniques of integration, such as substitution and partial fractions, is essential for solving integral problems.

Completing the Square

Completing the square is a method used to transform a quadratic expression into a perfect square trinomial. This technique is particularly useful in integration, as it simplifies the integrand, making it easier to evaluate. For the integral ∫ x/(x² + 6x + 18) dx, completing the square helps rewrite the denominator in a form that can be integrated using standard formulas.

Substitution Method

The substitution method is a technique used in integration to simplify the integrand by substituting a part of the expression with a new variable. This method is particularly effective when dealing with composite functions or when the integrand can be expressed in a simpler form. In the context of the given integral, choosing an appropriate substitution can lead to a more straightforward evaluation of the integral.