Question

Find the indefinite integral. (Hint: Integration by parts is not required for all the integrals.) $$ \int \frac{2 x}{e^{x}} d x $$

Short answer

The indefinite integral of \(\frac{2x}{e^{x}} dx\) is \(2Ei(-x) + C\).

Step-by-step solution

Recognize the integral form

The integral has the form \(\int u v dx\), where \(u = 2x\) and \(v = e^{-x}\). However, remember that integration by parts is not required for all the integrals. Examine the integral and see if there are any substitutions or simplifications that can be made before applying integration by parts.

Apply the substitution

Let's apply the substitution \(u=-x\). This leads to \(du=-dx\) or \(dx=-du\). Now the integral can be rewritten as \(- \int \frac{2(-u)}{e^{u}} du = 2 \int \frac{u}{e^{u}} du\)

Integrate using the Exponential Integral

The integral \(\int \frac{u}{e^{u}} du\) is a known integral represented as \(Ei(u)\) also known as the Exponential Integral.

Substitute back

Return the original variable \(x\) to place. The final solution will be \(2Ei(-x) + C\), where \(C\) is the constant of integration.

Key concepts

Integration by Parts

Integration by parts is a technique often used for integrating products of functions. It stems from the product rule of differentiation. The formula is:\[ \int u \, dv = uv - \int v \, du \]Here's how it works:

• Identify parts: Choose which function is \(u\) and which is \(dv\).
• Differential pieces: Differentiate \(u\) to find \(du\), and integrate \(dv\) to find \(v\).
• Substitute and solve: Plug the values into the formula and simplify.

In our exercise, integration by parts wasn't actually necessary, but it’s crucial to know when this technique is beneficial. It’s particularly useful for integrals like \( \int x e^x dx\). In solving problems, always check if another simpler method might be applicable before embarking on integration by parts.

Substitution Method

The substitution method transforms a complicated integral into a simpler one. It’s akin to reversing the chain rule of differentiation. Here’s the process:

• Choose substitution: Find a substitution \(u\) that simplifies the integral.
• Differentiate \(u\): Compute \(du\) in terms of \(dx\).
• Rewrite the integral: Substitute \(u\) and \(du\) into the integral, simplifying it.
• Integrate and back-substitute: Solve the simpler integral, and then convert back to the original variable by substituting again.

In the original exercise, substituting \(u = -x\) allowed us to rewrite the problem as an easier integral to solve. This transformation is instrumental in solving integrals with chains of functions or trigonometric substitutions.

Exponential Integral

The exponential integral \(Ei(x)\) is a special function used primarily in statistics and applied mathematics. Unlike basic functions, it doesn't have a simple expression in terms of elementary functions.Key aspects of \(Ei(x)\):

• Special function: Often defined through an integral, it's usually expressed or approximated via series or other functions.
• Usage: Commonly appears in solutions to certain types of integrals or differential equations.
• Not elementary: Requires lookup or computational methods to evaluate for specific values.

In our solution, the appearance of \(Ei(-x)\) was a typical result when attempting to integrate certain combinations of exponentials and polynomials. Always be aware when special functions like the exponential integral are involved, and consult additional resources if deeper understanding or computation is needed.