Problem 55 - Applied Calculus

Question

Find the indefinite integrals. $$\int\left(x^{2}+\frac{1}{x^{2}}\right) d x$$

Step-by-Step Solution

Answer

The indefinite integral is \(\frac{x^3}{3} - \frac{1}{x} + C\).

Step 1: Split the Integral

The given integral can be split into two separate integrals. This is because the integral of a sum of functions is the sum of their integrals. So, we rewrite the integral as:\[\int \left(x^{2} + \frac{1}{x^{2}}\right) dx = \int x^{2} dx + \int \frac{1}{x^{2}} dx\]

Step 2: Evaluate Each Integral Separately

We need to evaluate each part separately. Let's start with the first integral:\[\int x^{2} dx\]Apply the power rule for integration, which is \(\int x^n dx = \frac{x^{n+1}}{n+1} + C\). For \(x^2\), \(n = 2\), so:\[\int x^{2} dx = \frac{x^{3}}{3} + C_1\]

Step 3: Evaluate the Second Integral

Now, evaluate the second integral:\[\int \frac{1}{x^{2}} dx = \int x^{-2} dx\]Using the power rule again, with \(n = -2\):\[\int x^{-2} dx = \frac{x^{-1}}{-1} + C_2 = -\frac{1}{x} + C_2\]

Step 4: Combine the Integral Results

Now combine the results from Step 2 and Step 3. Remember, the constant of integration \(C\) can be a single constant, so you can combine \(C_1\) and \(C_2\) into just \(C\):\[\int \left(x^{2} + \frac{1}{x^{2}}\right) dx = \frac{x^{3}}{3} - \frac{1}{x} + C\]

Key Concepts

Power Rule for Integration

The power rule for integration is a fundamental principle used to solve integrals involving power functions. It states that for any real number \(n\), the integral of \(x^n\) with respect to \(x\) can be determined by the formula: \[ \int x^n \, dx = \frac{x^{n+1}}{n+1} + C \] where \(C\) is the constant of integration, added because the process of differentiation of a constant results in zero. This is valid for all \(n\) except \(-1\), as dividing by zero is undefined. When using the power rule, make sure to adjust the exponent (add 1 to it) and then divide by the new exponent. For example, when integrating \(x^2\), apply the rule as follows:

• The exponent \(n\) is 2, so add 1 to get 3.
• Divide the resultant expression by 3, giving \(\frac{x^3}{3}\).

The power rule simplifies many integration problems and is the go-to method for polynomials and other power functions.

Sum of Integrals

The sum of integrals property allows you to break down complex integrals involving sums into simpler, more manageable parts. This property states that the integral of a sum is equivalent to the sum of the integrals. Symbolically, if \(f(x)\) and \(g(x)\) are functions, then: \[ \int (f(x) + g(x)) \, dx = \int f(x) \, dx + \int g(x) \, dx \] This is particularly useful when dealing with problems like the one in the exercise, \(\int(x^2 + \frac{1}{x^2}) \, dx\). By separating the integral into two parts,

• \(\int x^2 \, dx\)
• \(\int \frac{1}{x^2} \, dx\)

we can apply different techniques to each to solve them individually. This approach simplifies the calculation process and aids in concentrating on one function at a time for more complex expressions.

Constant of Integration

The constant of integration, usually denoted as \(C\), is a crucial part of any indefinite integral. It represents any constant value that could have been differentiated away in the original function. Since differentiation of a constant yields zero, when we integrate back, this constant could have been any real number, thus we include \(C\) automatically. When integrating an expression, it's important to remember that each term might come with its own constant of integration, as seen in the solution.

• After integrating \(x^2\), you initially have \(\frac{x^3}{3} + C_1\).
• For \(\frac{1}{x^2}\), the result includes \(-\frac{1}{x} + C_2\).

To simplify, these constants \(C_1\) and \(C_2\) combine into a single constant \(C\), representing any possible number added to combine the results of both integrals. Including the constant of integration ensures all functions that lead to the same derivative are captured when integrating.