Question

Perform the indicated integrations. $$ \int \frac{x+1}{9 x^{2}+18 x+10} d x $$

Short answer

Perform partial fraction decomposition and then substitution to integrate.

Step-by-step solution

Simplify the Integrand

The first step is to simplify the integrand to make it easier to integrate. Notice that the denominator can be factored or the integrand can be simplified using algebraic techniques.

Factor the Quadratic in the Denominator

The quadratic expression in the denominator is \(9x^2 + 18x + 10\). Unfortunately, it does not factor neatly into simple binomials due to complex roots, so we proceed by completing the square or using partial fraction decomposition after making the simplification.

Perform Partial Fraction Decomposition

Rewrite the integrand by expressing the rational expression using partial fraction decomposition. The expression is already simple, but if needed, find constants \(A\) and \(B\) such that:\[\frac{x+1}{9x^2+18x+10} = \frac{Ax + B}{(3x + b)^2 + c}\] Though typically, use either substitution or recognize potential substitution patterns.

Make a Suitable Substitution

Recognize the square: \(9x^2+18x+10\) is rewritten (if simplified properly) and proceed to substitution:Let \( u = 9x^2+18x+10,\) then \(du = (18x+18)dx\).Adjusting for constants and form, solve for dx, and replace the integrand terms properly.

Simplify and Integrate

Utilize substitutions and factor manipulations to simplify the integral once in terms of \(u\) and solve the simplified integral using basic integration rules. A common approach might also leverage trig substitution for identical terms, or complete usual polynomial manipulations.

Evaluate and Replace Variables

Compute the integral you've set up in terms of \(u,\) and once you have the anti-derivative, replace the variable \(u\) back with its expression in terms of \(x,\) if applicable, to present the solution in the original variable.

Finalize the Solution and Simplify

Combine all parts logically to produce your concluding expression as you've integrated all parts. Apply known constants and confirm whether an integration constant is necessary.

Key concepts

Partial fraction decomposition

Partial fraction decomposition is a technique used in calculus to break down complex rational expressions into simpler fractions, which are easier to integrate. The basic idea is to express the fraction as a sum of simpler fractions, typically with linear or quadratic denominators. Here's a simple breakdown:

• First, ensure that the degree of the numerator is less than the degree of the denominator. If it's not, perform polynomial division.
• Factor the denominator into its irreducible components, if possible. If the quadratic has no real roots, consider completing the square.
• Express the original fraction as a sum of partial fractions. For each factor in the denominator, write a corresponding term(s) with unknown coefficients. Solve for these coefficients by clearing the fractions and equating coefficients.

This method simplifies the integration process, as it allows us to deal with each fraction independently. In cases where the denominator cannot be factored neatly, alternative methods like completing the square or substitution might be employed.

Polynomial integration

Polynomial integration involves finding the antiderivative of a polynomial function. This process is typically straightforward due to the power rule of integration. The power rule for integration states that for any term of the form \( ax^n \), its integral is \( \frac{a}{n+1}x^{n+1} + C \), where \( C \) is the constant of integration. Here are key steps:

• Apply the power rule individually to each term in the polynomial.
• Don't forget to add the constant of integration \( C \) after integrating.

For more complex expressions that involve fractions or require decomposing into partial fractions, you may need to first simplify the expression. But once in polynomial form, the power rule is your key tool. Remember, polynomials are the simplest type of function to integrate and a good starting point for developing more complex integration skills.

Substitution method

The substitution method is a powerful tool in calculus that transforms complex integrals into simpler ones by changing variables. It's often likened to the reverse of the chain rule in differentiation. Here's how to use it:

• Identify a substitution that simplifies the integral. Typically, you choose a part of the integrand whose derivative is also present.
• Let \( u \) be your substitution variable, often a function inside another function like \( g(x) \) in \( f(g(x)) \).
• Differentiate \( u \) with respect to \( x \) to find \( du \), and solve for \( dx \).
• Replace all \( x \) terms in the integral with expressions in \( u \), and integrate with respect to \( u \).
• After integration, replace \( u \) back with the original variable to obtain the solution.

This method is essential for integrals involving composed functions and is extremely useful when there's a clear inner function whose derivative is also present in the integrand.

Completing the square

Completing the square is a technique used to rewrite quadratic expressions in a form that is easier for integration or solving equations. It's particularly useful when dealing with quadratic functions that do not factor easily. Here's the approach:

• Write the quadratic expression \( ax^2 + bx + c \) in standard form.
• If \( a e 1 \), factor it out from the quadratic terms.
• Add and subtract the square of half the linear coefficient to/from the quadratic expression. This converts the quadratic into a perfect square trinomial plus a constant.
• Express the quadratic as \( a(x-h)^2 + k \), where \( h \) and \( k \) are derived from the above steps.

Once converted, the expression is much easier to integrate or analyze. Completing the square is particularly useful in deriving integrals or simplifying rational expressions that involve quadratics with no simple roots.