Question

\(\int\left[\left(e^{5 \log x}-e^{3 \log x}\right) /\left(e^{4 \log x}-e^{2 \log x}\right)\right] d x=\) (a) e \(\cdot 2^{-2 \mathrm{x}}\) (b) \(\mathrm{e}^{3} \log _{\mathrm{e}} \mathrm{x}\) (c) \(\left(\mathrm{x}^{3} / 3\right)\) (d) \(\left(\mathrm{x}^{2} / 2\right)\)

Short answer

\(\int\left[\left(e^{5 \log x}-e^{3 \log x}\right) /\left(e^{4 \log x}-e^{2\log x}\right)\right] dx = \left(\frac{x^2}{2}\right) + C\)

Step-by-step solution

Simplify the expression

We are given the expression: \(\int\left[\left(e^{5 \log x}-e^{3 \log x}\right) /\left(e^{4 \log x}-e^{2 \log x}\right)\right] dx\) Using the identity: \(e^{\log a^b} = a^b\), we can rewrite the expression as follows: \(\int\left[\left(x^5-x^3\right) / \left(x^4-x^2\right)\right] dx\)

Apply polynomial division

Let's simplify the expression using polynomial division, dividing the numerator by the denominator: \[\frac{x^5-x^3}{x^4-x^2} = x+\frac{x^3}{x^4-x^2}\] Now, our expression becomes: \(\int\left(x+\frac{x^3}{x^4-x^2}\right) dx\)

Separate integration terms

Next, we separate the integration of each term: \(\int x dx + \int \frac{x^3}{x^4-x^2} dx\)

Apply substitution

Now, let's integrate the second term using substitution. To do that, let's take \(u = x^2\), so \(du = 2x\, dx\). Next, we rewrite the second integral in terms of u: \(\int \frac{1/2\,u}{u^2-u} du\) Apply polynomial division for this term as well: \[\frac{1/2\,u}{u^2-u} = \frac{1}{2}\left( \frac{u}{u^2-u} \right) = \frac{1}{2}\left( \frac{1}{u-1} - \frac{1}{u} \right)\] Now, we will have: \(\int x dx + \int\left( \frac{1}{2}\left( \frac{1}{u-1} - \frac{1}{u} \right) \right) du\)

Integrate

Next, we will integrate each term: \[\int x dx = \frac{x^2}{2} + C_1\] \[\int\left( \frac{1}{2}\left( \frac{1}{u-1} - \frac{1}{u} \right) \right) du = \frac{1}{2}\int\frac{1}{u-1} du - \frac{1}{2}\int\frac{1}{u} du\] \[\frac{1}{2}\int\frac{1}{u-1} du = \frac{1}{2}\ln|u-1| + C_2\] \[-\frac{1}{2}\int\frac{1}{u} du = -\frac{1}{2}\ln|u| + C_3\]

Substitute back and combine terms

Now, we will substitute back \(u = x^2\) and combine the terms: \[\frac{x^2}{2} + \frac{1}{2}\ln|x^2-1| - \frac{1}{2}\ln|x^2| + C\]

Compare with the given options

Comparing the result with the given options, we can see that our integral matches with option (d): \(\boxed{\int\left[\left(e^{5 \log x}-e^{3 \log x}\right) /\left(e^{4 \log x}-e^{2\log x}\right)\right] dx = \left(\frac{x^2}{2}\right) + C}\)

Key concepts

Algebraic Manipulation

Algebraic manipulation is about rearranging and simplifying mathematical expressions to make them easier to understand or solve. In the context of our integral problem, algebraic manipulation includes applying known identities and simplifying the expression before beginning the integration process.

For our exercise, we start with the given integral expression and recognize we can simplify it using the identity \(e^{a \log b} = b^a\). This identity is a consequence of the exponent rules where an exponent raised to another exponent can be multiplied. Using this identity allows us to rewrite the integrand in terms of polynomial functions of \(x\), leading to an expression that is easier to handle. Effectively, through algebraic manipulation, we transform complex exponential forms into more familiar polynomials, setting the stage for further operations like polynomial division.

Polynomial Division

Polynomial division is akin to long division we use with numbers, but it applies to dividing polynomials. In the step-by-step solution provided, polynomial division is utilized to simplify the integral's numerator by its denominator. The process helps break down a complicated fraction into a simpler form or into a sum of simpler fractions.

In our specific exercise, after applying algebraic manipulation and exponent rules, we use polynomial division to simplify the integrand. The division of the polynomials \(x^5-x^3\) by \(x^4-x^2\) yields a quotient and possibly a remainder. This process results in a more straightforward expression for integration. Mastering polynomial division is essential for integration as it often converts an otherwise intimidating integral into a series of simpler ones that can be approached with basic integration techniques.

Logarithmic Identities

Logarithmic identities are rules that describe how logarithms behave under various mathematical operations. They are crucial in manipulating and solving equations involving logarithms. In the context of integration, these identities can help simplify an integral's integrand before attempting to solve it.

One such identity used in the solution is the property that \(e^{\text{log}(x^n)} = x^n\). This identity exploits the fact that the exponential function \(e^x\) and the natural logarithm \(\text{log}(x)\) are inverse operations. Simplifying the integral using logarithmic identities is a prime example of how understanding fundamental properties of mathematical functions can significantly reduce the complexity of a problem. Recognizing when and how to use logarithmic identities often reveals a path to an otherwise obscured solution.

Indefinite Integrals

Indefinite integrals represent the general form of antiderivatives; they include a constant of integration since an antiderivative is not unique. When solving indefinite integrals, it's about finding a function whose derivative gives back the original function within the integral sign. Common techniques include direct integration using known integral formulas, integration by parts, and integration by substitution.

In our solved exercise, we performed integration by substitution as a strategy to simplify the integral. We let \(u = x^2\) and rewrite parts of the integral in terms of \(u\), making it manageable to integrate. After integration, we substitute \(u\) back in terms of \(x\) to complete the solution. Grasping different techniques to solve indefinite integrals is key to tackling integration problems, as each approach offers a unique toolset to resolve complex situations that arise in calculus.