Question

$$\text {Evaluate the following integrals.}$$ $$\int \frac{x^{3}-10 x^{2}+27 x}{x^{2}-10 x+25} d x$$

Short answer

To evaluate the integral $$\int \frac{x^{3}-10 x^{2}+27 x}{x^{2}-10 x+25} d x,$$ we first perform polynomial long division and then integrate each term. The result is: $$\int \frac{x^{3}-10 x^{2}+27 x}{x^{2}-10 x+25} d x = \frac{x^2}{2} + 10x + 25 \ln|x^2 - 10x + 25| + \frac{1}{2} \ln|x^2 - 10x + 25| + C.$$

Step-by-step solution

Perform polynomial long division

Divide the numerator by the denominator to simplify the integrand: $$ \begin{array}{c|cc cc} \multicolumn{2}{r}{x^3} & -10x^2 & +27x \\ \cline{2-5} x^2 - 10x + 25 & x^3 & - 10x^2 & + 27x \\ \cline{2-4} \multicolumn{2}{r}{0} & 0 & - 27x & + 25 \\ \cline{4-5} \multicolumn{4}{r}{0} & 27x \\ \end{array} $$ So, we get the following result: $$ \int \frac{x^{3}-10 x^{2}+27 x}{x^{2}-10 x+25} d x = \int (x - 10 + \frac{27x - 25}{x^2 - 10x + 25}) d x $$

Integrate each term

Now, we will integrate each term in the resulting expression from the division. Namely, we will integrate \(x\), \(-10\), and separately the more complex term \(\frac{27x - 25}{x^2 - 10x + 25}\). $$ \int (x - 10 + \frac{27x - 25}{x^2 - 10x + 25}) d x = \int x d x - \int 10 d x + \int \frac{27x - 25}{x^2 - 10x + 25} d x $$

Integrate the simpler terms

The first two integrals are straightforward: $$ \int x dx = \frac{x^2}{2} + C_1 \quad \text{and} \quad \int 10 d x = 10x + C_2 $$

Integrate the complex term

The complex term in the integrand has the form of a derivative of a logarithm function, where \(u=x^2 - 10x + 25\). Notice that: $$\frac{d}{dx}(x^2-10x+25)=2x-10$$ Then, we can rewrite the complex term into the following expression: $$\int \frac{27x - 25}{x^2 - 10x + 25} d x = \int \frac{(27x - 25) + 25 - 2x}{x^2 - 10x + 25} d x = \int \frac{25 + (2x - 10)}{x^2 - 10x + 25} d x$$ Now, we can integrate this term: $$\int \frac{25 + (2x - 10)}{x^2 - 10x + 25} d x = 25 \underbrace{\int \frac{1}{x^2 - 10x + 25} d x}_{(*1)} + \frac{1}{2} \underbrace{\int \frac{2x - 10}{x^2 - 10x + 25} d x}_{(*2)}$$ For \((1)\), we perform a u-substitution. Let \(u = x^2 - 10x + 25\), then \(du = (2x - 10) dx\). Therefore, the integral becomes: $$\int \frac{1}{u}du = \ln |u|+C_3=\ln|x^2-10x+25|+C_3$$ For \((2)\), since it is in the form of \(\int \frac{U'(x)}{U(x)} dx\), the integral is direct: $$\frac{1}{2}\int \frac{2x - 10}{x^2 - 10x + 25} d x = \frac{1}{2} \ln|x^2 - 10x + 25| + C_4$$

Combine all terms and write the final result

Finally, we will combine all terms obtained from the integrals: $$ \int (x - 10 + \frac{27x - 25}{x^{2}-10 x+25}) d x = \frac{x^2}{2} + 10x + 25 \ln|x^2 - 10x + 25| + \frac{1}{2} \ln|x^2 - 10x + 25| + C $$ Where \(C = C_1+C_2+C_3+C_4\) is the constant of integration.

Key concepts

Polynomial Long Division

Polynomial long division is a technique similar to numerical long division but applied to polynomials. It's used here to simplify the integrand, making complicated fractions easier to integrate. The first step is to divide the numerator polynomial by the denominator.
Here's how it generally works:

• Identify the leading term of both the numerator and the denominator.
• Divide the leading term of the numerator by the leading term of the denominator.
• Multiply the entire denominator by this result and subtract it from the numerator.
• Repeat the process with the new polynomial until the degree of the remainder is less than the degree of the denominator.

In this particular problem, the process simplifies the original expression to: \[ \int (x - 10 + \frac{27x - 25}{x^2 - 10x + 25}) \, dx \]This simplification is crucial as it breaks down the problem into manageable parts, making the integration of complex rational expressions more straightforward.

U-substitution

U-substitution is a valuable technique in calculus used to simplify the process of integration. It involves substituting part of the integral with a new variable, usually denoted as \(u\), making the integral easier to solve.
Here's how it works:

• Identify a portion of the integral that can be set as \( u \).
• Find the derivative \( du \) in terms of the original variable.
• Substitute and replace all occurrences of the chosen part and the differential to transform the integral into a basic form.

In our example, we let \( u = x^2 - 10x + 25 \) with \( du = (2x - 10)\, dx \). This transforms some integrals into a natural logarithmic form:\[ \int \frac{1}{u} \, du = \ln|u| + C \]This method simplifies what might initially seem like a daunting integral, turning it into a straightforward logarithmic integration.

Logarithmic Integration

Logarithmic integration deals with integrals that can simplify into a logarithmic form, often through techniques like u-substitution.
Here's why logarithms play a crucial role:

• Logarithmic integration is often used when an integral resembles the derivative of a logarithmic function.
• The basic rule used is: \( \int \frac{1}{x} \, dx = \ln|x| + C \).
• Recognizing these forms helps in simplifying complex rational integrals efficiently.

In this exercise, after employing polynomial long division and u-substitution, the integral:
\[ \int \frac{25 + (2x - 10)}{x^2 - 10x + 25} \, dx \]
was broken down into simpler terms, which include logarithmic forms like:
\[ 25 \ln|x^2 - 10x + 25| + \frac{1}{2} \ln|x^2 - 10x + 25| \]By transforming unwieldy algebraic expressions into logarithmic terms, the integration process becomes accessible and clear.