Problem 6 - Essential Calculus

Question

Find the integral. $$ \int \frac{x^{4}-1}{x^{2}+1} d x $$

Step-by-Step Solution

Answer

The solution to the integral \( \int \frac{x^{4}-1}{x^{2}+1} \, dx \) is \( \frac{1}{3}x^3 - x - \arctan(x) + C \)

Step 1: Divide each term separately

Divide each term in the numerator by the denominator. We get: \( \frac{x^4}{x^2 + 1} - \frac{1}{x^2 + 1} \)

Step 2: Simplify

Simplify the first term to \( x^2 - 1 \). The simplified version of the integral will be: \( \int (x^2 - 1 - \frac{1}{x^2 + 1}) \, dx \)

Step 3: Integrate

The integrals are calculated separately as:The integral of \( x^2 \) is \( \frac{1}{3}x^3 \), of \( 1 \) is \( x \) and \( \frac{1}{x^2 + 1} \) is \( \arctan(x) \). So, we get: \( \frac{1}{3}x^3 - x - \arctan(x) + C \) where \( C \) is the constant of integration.

Key Concepts

Integration Techniques

Integration techniques are crucial in calculus to find functions given their derivative, commonly known as integrals. They help us calculate the area under a curve, solve differential equations, and much more. In this problem, we need to find the integral of a rational function: \[ \int \frac{x^{4}-1}{x^{2}+1} \, dx \]Various integration techniques can be applied, such as:

• Substitution: Useful when the integrand features a function and its derivative.
• Partial Fractions: Effective when dealing with rational functions with factors in the denominator.
• By Parts: This technique is useful when the integrand is a product of two functions.

Each of these methods has its scenarios where it's most effective. In our exercise, we did not directly apply these techniques but instead simplified the integral using polynomial long division before integrating term by term.

Polynomial Long Division

Polynomial long division is similar to long division with numbers and is often used to simplify integrals involving polynomials. It helps by dividing the degree of the polynomial in the numerator by the degree of the polynomial in the denominator. In our exercise, we performed the polynomial division to rewrite:\[ \frac{x^{4}-1}{x^{2}+1} \]This was done by dividing each term separately, resulting in:\[ \frac{x^4}{x^2 + 1} - \frac{1}{x^2 + 1} \]After simplification, the rational term turned into a simpler polynomial expression, which was more straightforward to integrate: \[ x^2 - 1 - \frac{1}{x^2 + 1} \]This division method is handy when you need to deal with high-degree polynomials or decompose complex rational expressions into simpler terms. It makes the integration process much more manageable.

Trigonometric Integrals

Trigonometric integrals often involve trigonometric functions in the integrand, and specific methods make their integration more straightforward. In exercises like our present case, we encounter the inverse trigonometric functions for integration.In our example, we had a term:\[ \frac{1}{x^2 + 1} \]The integral of this specific function is known to be:\[ \arctan(x) \]This is because the derivative of \( \arctan(x) \) is \( \frac{1}{x^2 + 1} \). The familiarity with inverse trigonometric integrals like this one is essential because similar forms appear frequently within calculus problems.Understanding how to handle these trigonometric integrals aids in recognizing situations where substitution or recognition of standard results can simplify your work significantly. It's fundamental to learn common trigonometric integration results to speed up problem-solving across different calculus problems.