Problem 36 - Calculus Single Variable

Question

Evaluate the given definite integral. \(\int_{1}^{2} \frac{7 x^{2}+6}{x^{2}\left(x^{2}+1\right)} d x\)

Step-by-Step Solution

Answer

\(\ln 2 + 3 + \frac{\pi}{4} - \tan^{-1}(2)\) is the evaluated integral.

Step 1: Simplify the Integral Expression

The integral to solve is \(\int_{1}^{2} \frac{7x^2 + 6}{x^2(x^2+1)} \, dx\). Begin by simplifying the expression. Divide both the numerator \(7x^2 + 6\) by \(x^2\), getting it into a form that's easier to integrate: \(\frac{7x^2 + 6}{x^4 + x^2}\). Yet, it can be decomposed using partial fraction decomposition.

Step 2: Apply Partial Fraction Decomposition

Express the integrand \(\frac{7x^2 + 6}{x^2(x^2+1)}\) as a sum of simpler fractions. Start by assuming:\[\frac{A}{x} + \frac{B}{x^2} + \frac{Cx + D}{x^2 + 1}\]. Multiply through by the common denominator \(x^2(x^2+1)\) to get: \[7x^2 + 6 = A x(x^2+1) + B(x^2+1) + (Cx + D)x^2\].Expand and collect like terms.

Step 3: Solve the System of Equations

Compare the coefficients of powers of \(x\) from both sides:1. For \(x^3\): \(A + C = 0\)2. For \(x^2\): \(A + B + D = 7\)3. For \(x^1\): \(C = 0\)4. For the constant term: \(B = 6\)Solving, \(C = 0\), \(B = 6\), \(A + D = 7\), and \(A = 0\) (as C is zero and doesn't contribute to other terms). Thus, \(A = 1\) and \(D = 1\). So:\[\frac{1}{x} + \frac{6}{x^2} + \frac{1}{x^2+1}\].

Step 4: Integrate Each Term Separately

The integral becomes three separate integrals: \[ \int_{1}^{2} \frac{1}{x} \, dx + \int_{1}^{2} \frac{6}{x^2} \, dx + \int_{1}^{2} \frac{1}{x^2+1} \, dx\].Integrate each:1. \(\int \frac{1}{x} \, dx = \ln|x| + C\)2. \(\int \frac{6}{x^2} \, dx = -\frac{6}{x} + C\)3. \(\int \frac{1}{x^2+1} \, dx = \tan^{-1}(x) + C\).Evaluate each from 1 to 2.

Step 5: Evaluate the Integrals with Limits

Compute the definite integrals:1. \(\left[ \ln|x| \right]_{1}^{2} = \ln(2) - \ln(1) = \ln 2\)2. \(\left[ -\frac{6}{x} \right]_{1}^{2} = -3 + 6 = 3\)3. \(\left[ \tan^{-1}(x) \right]_{1}^{2} = \tan^{-1}(2) - \tan^{-1}(1) = \frac{\pi}{4} - \tan^{-1}(2)\).Combining:\(\ln 2 + 3 + \frac{\pi}{4} - \tan^{-1}(2)\). The arithmetic completes with irrational constants.

Key Concepts

Partial Fraction Decomposition

Partial Fraction Decomposition is a technique used to simplify complicated rational expressions into simpler fractions that are easier to work with, especially in integration. When an expression is decomposed into more manageable parts, calculating an integral becomes much straightforward. To apply this method, you express the given function as a sum of simpler fractions based on its denominator’s factors. For the case of

• \( \frac{7x^2 + 6}{x^2(x^2+1)} \)

Assume the expression can be written as

• \( \frac{A}{x} + \frac{B}{x^2} + \frac{Cx + D}{x^2 + 1} \)

By equating the simplified terms and matching coefficients for corresponding powers of \(x\), you can solve for constants \(A\), \(B\), \(C\), and \(D\). This decomposition enables each fraction to be integrated separately, simplifying the overall integration task.

Integral Evaluation

Integral Evaluation involves calculating the area under a curve described by a function over a specific interval. The definite integral has limits of integration bound by two values, which in this exercise are from 1 to 2. To evaluate

• \( \int_{1}^{2} \frac{7x^2 + 6}{x^2(x^2+1)} \, dx \)

You need to first decompose it into simpler parts, achieved through Partial Fraction Decomposition. After obtaining simpler components, evaluate the integrals individually within the specified limits.
Here's the breakdown for this task:

• \( \int_{1}^{2} \frac{1}{x} \, dx \)
• \( \int_{1}^{2} \frac{6}{x^2} \, dx \)
• \( \int_{1}^{2} \frac{1}{x^2+1} \, dx \)

Each term contributes specifically to the final result, showcasing individual evaluation's importance for definite integrals.

Calculus

Calculus is the branch of mathematics that studies how things change. Its two main focuses are differentiation and integration. Integration is about summing parts to find the whole, such as areas under curves, which is what this problem involves.
In calculus, definite integrals are used to compute exact areas, volumes, and more. In our example

• \( \int_{1}^{2} \frac{7x^2 + 6}{x^2(x^2+1)} \, dx \)

The integration process ties together aspects like simplification and decomposing complex functions, making it easier to visualize the definite area under a curve between two specific values. When individual parts are integrated and boundaries applied, we acquire precise values, revealing integral solutions' power.

Integration Techniques

Integration Techniques in calculus are varied, and choosing the right one can simplify problems significantly. The exercise at hand required the use of Partial Fraction Decomposition as a technique to handle complex rational expressions. This solution strategy makes it feasible to calculate integrals of convoluted functions by turning them into simpler terms.
The basic techniques you can use in integration include:

• Substitution: Used when a change of variables makes the integral simpler.
• Integration by parts: Useful for products of functions.
• Partial Fraction Decomposition: Especially good for rational functions.
• Trigonometric integration: Helpful when integrating functions involving trigonometric terms.

By applying Partial Fraction Decomposition effectively, expressions were broken down, streamlining the integral evaluation. Each step becomes intuitive once you familiarize yourself with suitable techniques, transforming otherwise complex problems into more solvable formats.