Question

Evaluate the following integrals. $$\int \frac{-x^{5}-x^{4}-2 x^{3}+4 x+3}{x^{2}+x+1} d x$$

Short answer

Following the step-by-step solution, the final result for the integral is: $$\int \frac{-x^{5}-x^{4}-2 x^{3}+4 x+3}{x^{2}+x+1} dx = -\frac{1}{4}x^4 +\frac{1}{3}x^3 +\frac{1}{2}x^2 -3\ln|x+\frac{1}{2} - \frac{1}{2}|+3\ln|x+\frac{1}{2} + \frac{1}{2}|+C$$

Step-by-step solution

Long Division

First, simplify the integrand with long division. Perform the division: $$ \begin{array}{c|cc ccc} \multicolumn{2}{r}{-x^3} & -x^2 & -x \\ \cline{2-6} x^2+x+1 & -x^5 & -x^4 & -2x^3 & +4x & +3 \\ \cline{2-4} \multicolumn{2}{r}{x^5} & +x^4 & +x^3 \\ \cline{2-4} \multicolumn{2}{r}{0} & 0 & -x^3 & -1x \\ \cline{4-5} \multicolumn{4}{r}{x^3} & +x^2 & +x \\ \cline{4-6} \multicolumn{4}{r}{0} & 0 & -3x & +3\\ \cline{6-7} \multicolumn{6}{r}{3x} & +3 \\ \cline{6-7} \multicolumn{6}{r}{0} & 0 \\ \end{array} $$ So, after the long division, we have found: $$\frac{-x^5-x^4-2x^3+4x+3}{x^2+x+1} = -x^3-x^2-x+\frac{-3x+3}{x^2+x+1}.$$

Integrate Term by Term

Now, integrate the simplified expression with respect to x. Split the integral into a sum of integrals: $$\int (-x^3-x^2-x+\frac{-3x+3}{x^2+x+1}) dx = \int -x^3 dx - \int x^2 dx - \int x dx + \int \frac{-3x+3}{x^2+x+1} dx.$$

Evaluate Elementary Integrals

Evaluate the elementary integrals: $$\int -x^3 dx = -\frac{1}{4}x^4 +C_1$$ $$\int x^2 dx = \frac{1}{3}x^3+C_2$$ $$\int x dx = \frac{1}{2}x^2 + C_3$$ Note that the indefinite integral that is left, \(\int \frac{-3x+3}{x^2+x+1}dx\), is more complex. We'll handle that in the remaining steps.

Integrate the More Complex Term using a substitution

To evaluate the last integral \(\int \frac{-3x+3}{x^2+x+1}dx\), let's use the substitution method. We know that: $$\int \frac{-3x+3}{x^2+x+1}dx = \int \frac{-3(x+\frac{1}{2})+3}{x^2+x+1}dx.$$ Let \(u=x+\frac{1}{2} \rightarrow x=u-\frac{1}{2}\); then \(dx=du\). The integral now becomes: $$\int \frac{-3u+3}{u^2-\frac{1}{4}}du.$$

Integrate Using Partial Fraction Decomposition

To evaluate the integral, we can use partial fraction decomposition: $$\frac{-3u+3}{u^2-\frac{1}{4}}=\frac{A}{u-\frac{1}{2}}+\frac{B}{u+\frac{1}{2}}$$ Clear of denominator: $$-3u+3=(A+B)u^2+(A-B)\frac{1}{2}$$ Matching coefficients, we get: $$A+B=0$$ $$A-B=-6$$ By solving these linear equations, we find that \(A = -3\) and \(B = 3.\) The integral now becomes: $$\int\frac{-3}{u-\frac{1}{2}}du + \int\frac{3}{u+\frac{1}{2}}du$$

Evaluate the Last Integrals

Now, evaluate the last two integrals: $$\int\frac{-3}{u-\frac{1}{2}}du = -3\ln|u-\frac{1}{2}|+C_4$$ $$\int\frac{3}{u+\frac{1}{2}}du = 3\ln|u+\frac{1}{2}|+C_5$$ Substitute back \(u=x+\frac{1}{2}\) to get: $$-3\ln|x+\frac{1}{2} - \frac{1}{2}|+3\ln|x+\frac{1}{2} + \frac{1}{2}|$$

Combine All the Integrals

Now, we can combine all the constants into a single constant \(C= C_1+C_2+C_3+C_4+C_5\), and the final integral result is: $$\int \frac{-x^{5}-x^{4}-2 x^{3}+4 x+3}{x^{2}+x+1} dx = -\frac{1}{4}x^4 +\frac{1}{3}x^3 +\frac{1}{2}x^2 -3\ln|x+\frac{1}{2} - \frac{1}{2}|+3\ln|x+\frac{1}{2} + \frac{1}{2}|+C$$

Key concepts

Long Division in Integration

Long division is a technique often used in integration to simplify the polynomial division when dealing with complex rational functions. It works in a similar manner to the traditional long division method you might remember from arithmetic, but it is applied to polynomials. In the context of integration, this process is used to break down a complex rational function into simpler parts which are easier to integrate.

To perform polynomial long division, follow these general steps:

• Divide the leading term of the dividend by the leading term of the divisor.
• Multiply the entire divisor by the result obtained in the previous step.
• Subtract this product from the dividend.
• Repeat the process with the resulting polynomial until the degree of the remainder is less than the degree of the divisor.

This simplification through long division often leads to a polynomial and a simpler rational expression, making the integration process smoother. For example, in our problem, the division of \(-x^5-x^4-2x^3+4x+3\) by \(x^2+x+1\) gives us the simpler expression \(-x^3-x^2-x+\frac{-3x+3}{x^2+x+1}\), where the rational expression can be tackled with other methods in integration.

Substitution Method

The substitution method is a powerful tool in calculus that helps to simplify integrals by changing the variable of integration. It works on the principle of making a substitution that simplifies the integral into a more familiar or manageable form.

When using substitution, the process involves:

• Selecting a substitution, which is usually part of the integrand, that complicates the integral.
• Expressing \(dx\) in terms of the new variable.
• Rewriting the integral in terms of the new variable.
• Integrating the new expression with respect to the substituted variable.
• Substituting back to the original variable once integration is complete.

In the provided exercise, the substitution \(u = x + \frac{1}{2}\) is used to simplify the integral \(\int \frac{-3x+3}{x^2+x+1} dx\). The derivative \(dx\) becomes \(du\), turning the integral into one in \(u\), effectively simplifying the evaluation process.

Partial Fraction Decomposition

Partial Fraction Decomposition is a technique utilized to break down complex rational expressions into simpler fractions that are easier to integrate. This method works specifically well when the integrand is a rational function, which means it's a ratio of two polynomials.

Here's how you generally perform partial fraction decomposition:

• Ensure the degree of the numerator is less than the degree of the denominator. If not, use long division first.
• Factor the denominator completely into linear and/or irreducible quadratic factors.
• Express the integrand as a sum of partial fractions, assigning constants to unknown coefficients, which represent the numerators of the fractional components.
• Solve for these constants using algebraic techniques such as equating coefficients.

In this exercise, after substituting \(u\), the integral \(\int \frac{-3u+3}{u^2-\frac{1}{4}} du\) was decomposed into partial fractions: \(\frac{-3}{u-\frac{1}{2}} + \frac{3}{u+\frac{1}{2}}\). This decomposition allowed for straightforward integration of each term, transforming the difficult expression into individually manageable parts.