Problem 41 - Thomas Calculus

Question

Use integration by parts to establish the reduction formula. \(\int x^{n} e^{a x} d x=\frac{x^{n} e^{a x}}{a}-\frac{n}{a} \int x^{n-1} e^{a x} d x, \quad a \neq 0\)

Step-by-Step Solution

Answer

Apply integration by parts with \( u = x^n \) and \( dv = e^{ax} \, dx \), simplify to get the given reduction formula.

Step 1: Choose Functions for Integration by Parts

Integration by parts is given by \( \int u \, dv = uv - \int v \, du \). Here, let us choose \( u = x^n \) and \( dv = e^{ax} \, dx \). This choice will make the differentiation of \( u \) simple, helping us simplify the integral.

Step 2: Differentiate u and Integrate dv

Differentiate \( u = x^n \) to get \( du = n x^{n-1} \, dx \).Integrate \( dv = e^{ax} \, dx \) to obtain \( v = \frac{e^{ax}}{a} \).

Step 3: Apply the Integration by Parts Formula

Substitute \( u \), \( v \), \( du \), and \( dv \) into the integration by parts formula. This gives us:\[\int x^n e^{ax} \, dx = \frac{x^n e^{ax}}{a} - \int \frac{e^{ax}}{a} \cdot n x^{n-1} \, dx\]

Step 4: Simplify the Resulting Integral

The expression for the remaining integral is:\[- \frac{n}{a} \int x^{n-1} e^{ax} \, dx \]This matches the second term on the right side of our reduction formula, completing the derivation.

Key Concepts

Integration by Parts

Integration by parts is a powerful technique used in integral calculus to integrate products of functions. It is derived from the product rule for differentiation and is expressed as:\[ \int u \, dv = uv - \int v \, du \]In this formula, we cleverly choose one part of the integrand as \( u \) (which we differentiate) and the other part as \( dv \) (which we integrate). The goal is to simplify the integral into a more manageable form.

• Choose \( u \) such that its differentiation simplifies the expression.
• Choose \( dv \) so that you can easily integrate it.

By applying this strategy, we can establish a reduction formula. This transformation reduces the complexity of the integral each step until a trivial solution is reached. The process of building a reduction formula emphasizes not only technique but also insight into the nature of the functions involved.

Exponential Functions

Exponential functions are a crucial part of calculus, especially when dealing with integration and differentiation. They have the general form \( e^{ax} \), where \( e \) is the base of the natural logarithm, approximately equal to 2.718. These functions have unique properties:

• They grow or decay at a rate proportional to their current value.
• The derivative and integral of an exponential function are proportional to the function itself.

When we integrate or differentiate the function \( e^{ax} \), it maintains its form. During integration by parts, choosing an exponential function as \( dv \) allows us to leverage its predictable behavior and simplify the integral. For instance, integrating \( e^{ax} \, dx \) yields \( \frac{e^{ax}}{a} \), which is simple and leads to efficient solutions in subsequent calculations.

Differentiation

Differentiation is a fundamental concept in calculus, focusing on finding the derivative of a function. The derivative represents the rate at which a function changes at any given point, analogous to finding the slope of a curve at a particular point. For the function \( u = x^n \), differentiating it involves applying the power rule:\[ \frac{d}{dx}(x^n) = nx^{n-1} \]This power rule is straightforward and widely used because it simplifies expressions by reducing the power of \( x \) by one. In the integration by parts technique, differentiation of \( u \) produces \( du \), which contributes to constructing a new integral that is often simpler to solve. By transforming expressions like \( x^n \) to \( nx^{n-1} \, dx \), we methodically break down complex integrals into simpler parts.

Integral Calculus

Integral calculus is dedicated to finding the total accumulation of quantities and areas under curves. The opposite of differentiation, it seeks to reverse the process of finding derivatives. Integration can appear challenging, especially for non-trivial functions, hence the need for techniques like integration by parts.

• Definite integrals calculate the exact area under a curve between two points.
• Indefinite integrals provide a family of functions as solutions, including an arbitrary constant \( C \).

In practice, integral calculus is used in various fields including physics, engineering, and economics to solve real-world problems involving continuous change. The reduction formula derived through integration by parts simplifies the process of integrating complex functions like \( x^n e^{ax} \). By repeatedly applying this formula, we can break down the original integral into manageable steps, ultimately achieving a solution.