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Chapter 30: Problem 12

Determine the following: $$ \int \frac{3 x^{3}-4 x^{2}+3 x}{x^{2}+1} d x $$

Short Answer

Expert verified
The short answer for the integral is: $$ \int \frac{3 x^{3}-4 x^{2}+3 x}{x^{2}+1} d x = \frac{3}{2}x^2 - 4x + \frac{11}{2} \ln |x^2 + 1| + 4 \arctan(x) + C $$

Step by step solution

01

Perform long division

First, we will perform long division on the given function. We want to divide the numerator by the denominator: $$ \frac{3x^3 - 4x^2 + 3x}{x^2 + 1} $$ Performing long division, we obtain: $$ 3x - 4 + \frac{11x - 4}{x^2 + 1} $$
02

Integrate term by term

Now that we have rewritten the integrand, we can integrate term by term: $$ \int \left( 3x - 4 + \frac{11x - 4}{x^2 + 1} \right) dx = \int 3x dx - \int 4 dx + \int \frac{11x - 4}{x^2 + 1} dx $$
03

Integrate the first two terms

Integrating the first two terms is straightforward: $$ \int 3x dx = \frac{3}{2}x^2 + C_1 $$ and $$ \int 4 dx = 4x + C_2 $$
04

Integrate the third term

To integrate the third term, we first split it into two simpler fractions: $$ \int \frac{11x - 4}{x^2 + 1} dx = \int \frac{11x}{x^2 + 1} dx - \int \frac{4}{x^2 + 1} dx $$ Now, we make the following substitution for the first integral: Let \(u = x^2 + 1\), then \(du = 2x dx\). We have, $$ \int \frac{11x}{x^2 + 1} dx = \frac{11}{2} \int \frac{1}{u} du $$ Integrating with respect to \(u\), we obtain: $$ \frac{11}{2} \int \frac{1}{u} du = \frac{11}{2} \ln |u| + C_3 = \frac{11}{2} \ln |x^2 + 1| + C_3 $$ Next, we integrate the second part: $$ \int \frac{4}{x^2 + 1} dx = 4\int \frac{1}{x^2 + 1} dx = 4 \arctan(x) + C_4 $$
05

Combine all the components

Finally, we combine all the components and simplify the result: $$ \int \frac{3 x^{3}-4 x^{2}+3 x}{x^{2}+1} d x = \frac{3}{2}x^2 - 4x + \frac{11}{2} \ln |x^2 + 1| + 4 \arctan(x) + C $$ where C is the combined constant (C = C_1 + C_2 + C_3 + C_4).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Long Division in Calculus
Long division can be a handy tool in calculus when working with polynomial fractions. It's similar to the long division you learn in arithmetic, but used here to simplify complex fractions before integration.
To apply long division in calculus, follow these steps:
  • Divide the leading term of the numerator by the leading term of the denominator.
  • Multiply the entire divisor by the result and subtract from the numerator.
  • Repeat these steps until the degree of the remainder term is less than the degree of the divisor.
In our problem, we divided \(3x^3 - 4x^2 + 3x\) by \(x^2 + 1\), resulting in \(3x - 4\) with a remainder of \(11x - 4\).
This simplifies the integrand and makes the integration process more manageable.
Integration Techniques
Integration techniques are essential for solving integrals that do not fit the basic formulas. Here, once long division was performed, we integrated each term separately using basic integration rules.
For the function \(3x - 4 + \frac{11x - 4}{x^2 + 1}\), the integration proceeds as follows:
  • For \(3x\), use the power rule: \(\int 3x \, dx = \frac{3}{2}x^2\).
  • For constant \(-4\), integrate straightforwardly: \(\int -4 \, dx = -4x\).
Finally, each piece can be combined to provide simpler sub-integrals, paving the way to integrate complicated expressions effectively.
Substitution Method
The substitution method is a technique to simplify integrals, particularly when dealing with a composite function. It's akin to reverse engineering a chain rule of differentiation.
In our example, we used substitution to better handle the integral \(\int \frac{11x}{x^2 + 1} \, dx\). Here, the substitution \(u = x^2 + 1\) was made with \(du = 2x \, dx\).
This transformed our integral into more manageable terms: \(\frac{11}{2} \int \frac{1}{u} \, du\), which integrates to \(\frac{11}{2} \ln |u|\) or \(\frac{11}{2} \ln |x^2 + 1|\) when back-substituting \(u\).
Substitution can effectively simplify problems that seem complex at first glance.

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