Question

\(\int\left[(x+1) d x /\left\\{x\left(1+x e^{x}\right)^{2}\right\\}\right]=\) (a) \(\log \left(\mathrm{xe}^{\mathrm{x}}\right) /\left(1+\mathrm{xe}^{\mathrm{x}}\right) \mid-\left[1 /\left(1+\mathrm{xe}^{\mathrm{x}}\right)\right]\) (b) \(\log \left|\left(x e^{x}\right) /\left(1+x e^{x}\right)\right|-\left[1 /\left(1+x e^{x}\right)\right]\) (c) \(\log \left(\mathrm{xe}^{\mathrm{x}}\right) /\left(1+\mathrm{xe}^{\mathrm{x}}\right) \mid+\left[1 /\left(1+\mathrm{xe}^{\mathrm{x}}\right)\right]\) (d) \(\log \left|\left(1+x e^{x}\right) /\left(x e^{x}\right)\right|-\left[1 /\left(1+x e^{x}\right)\right]\)

Short answer

\[-\ln \left|\frac{(xe^x)}{(1+xe^x)}\right| + \frac{1}{(1+xe^x)}\]

Step-by-step solution

Substitution Method

To start with, we use the substitution method. We let: \[u=1+xe^x\] Now, differentiate u with respect to x: \[du = (0+e^x + xe^x)dx\] Since \(x+1 = u - e^x\), we can replace the numerator: \[dx = \frac{du}{e^x + xe^x}\] Now we can rewrite the integral using the substitution: \[\int \frac{(x+1)}{x(1+xe^x)^2} dx = \int \frac{u - e^x}{xu^2} \frac{du}{e^x + xe^x}\]

Simplify the Integral

We can observe that \(e^x\) can be factored out in the denominator of the new integral to simplify it: \[\int \frac{u - e^x}{xu^2} \frac{du}{e^x + xe^x} = \int \frac{u - e^x}{xu^2} \frac{du}{e^x(1+x)}\] Now, the terms \(x(1+x)e^x\) in the denominator get canceled: \[\int \frac{u - e^x}{xu^2} \frac{du}{e^x(1+x)} = \int \frac{u - e^x}{u^2} du\]

Split the Integral

Now, let's split the integral into two parts: \[ \int \frac{u - e^x}{u^2} du = \int \frac{u}{u^2} du - \int \frac{e^x}{u^2} du\]

Solve the Integral

Now we can solve the remaining individual integrals: \[ \int \frac{u}{u^2} du - \int \frac{e^x}{u^2} du = - \int u^{-1} du - \int \frac{e^x}{(1+xe^x)^2}du\] The first integral gives: \[ - \int u^{-1} du = -\ln{|u|} + C_1\] And for the second integral, we shall substitute back \(u = 1+xe^x\) and solve: \[- \int \frac{e^x}{(1+xe^x)^2}du = -\int \frac{1}{u^2}du = \frac{1}{u} + C_2\]

Combine the Integrals and Rewrite

Now combine the two integrals' solutions and substitute back the original value of \(u\): \[-\ln{|u|} + \frac{1}{u} + C = -\ln{(1+xe^x)} + \frac{1}{1+xe^x} + C\] Comparing to the given options, this matches option (b). Therefore, our final answer is: \[-\ln \left|\frac{(xe^x)}{(1+xe^x)}\right| + \frac{1}{(1+xe^x)}\]

Key concepts

Indefinite Integral

An indefinite integral represents a family of functions whose derivatives are given by the integrand. It is often notated with the integral sign and a differential, such as \( \int f(x)dx \), without boundary limits. The result includes a constant term (\( C \)), the so-called constant of integration, because the derivative of a constant is zero and therefore has no effect on differentiation. Understanding the indefinite integral is crucial, as it allows us to reverse the process of differentiation, finding the original functions—referred to as antiderivatives—that when differentiated, yield the function under the integral sign.

When faced with a function that is difficult to integrate directly, as shown in our exercise, integration by substitution can simplify the process. The exercise involves a rational function with an exponential, which leads us to choose an appropriate substitution to reduce the integral to a more manageable form.

Integration Techniques

There are several integration techniques one can use to calculate indefinite integrals, one of which is substitution, as demonstrated in the solution. The goal of substitution—often referred to as u-substitution—is to simplify the integral by transforming it into one that is easier to evaluate. It involves choosing a part of the integral to replace with a variable (commonly \( u \)), such that the integral is expressed in terms of \( u \), and then finding the differential \( du \).

Once the integral is simplified through substitution, we may have to use additional techniques to proceed. This may include algebraic manipulation, such as factoring or expanding to reduce complex expressions into simpler terms, or splitting the integral into more than one part, each of which can be integrated separately. These strategies, along with substitution, turn what might at first seem like an intractable integral into a series of more straightforward calculations.

Exponential Functions

An exponential function is a mathematical function of the form \( f(x) = e^{x} \), where \( e \) is Euler's number, approximately equal to 2.71828. This type of function is known for its unique property where the rate of growth (or decay, if the exponent is negative) is proportional to its value. Exponential functions appear frequently in various domains such as economics, physics, and biology.

In the context of integration, exponential functions are interesting because their antiderivatives maintain the same form. For example, the integral of \( e^{x} \) with respect to \( x \) is \( e^{x} + C \), where \( C \) is the constant of integration. Their integration often leads to a simplification when combined with certain types of functions, as seen in the exercise where the presence of an exponential function within a more complex expression allows for the application of substitution techniques to greatly simplify the problem. In summary, recognizing the presence of an exponential function within an integral can be a crucial step towards finding its antiderivative.