Question

apply integration by parts twice to evaluate each integral. $$ \int x^{2} e^{x} d x $$

Short answer

\( \int x^2 e^x \, dx = e^x (x^2 - 2x + 2) + C \).

Step-by-step solution

Choose Functions for Integration by Parts

Integration by parts formula is \( \int u \, dv = uv - \int v \, du \). We choose \( u = x^2 \) and \( dv = e^x \, dx \). This makes \( du = 2x \, dx \) and \( v = e^x \), because the derivative of \( x^2 \) is \( 2x \) and the integral of \( e^x \) is \( e^x \).

Apply Integration by Parts (First Application)

Substitute into the integration by parts formula: \( \int x^2 e^x \, dx = x^2 e^x - \int e^x (2x) \, dx \). This simplifies to \( x^2 e^x - 2 \int x e^x \, dx \).

Choose Functions for Second Application

For the integral \( \int x e^x \, dx \), choose \( u = x \) and \( dv = e^x \). Then \( du = dx \) and \( v = e^x \).

Apply Integration by Parts (Second Application)

Substitute into the integration by parts formula again: \( \int x e^x \, dx = x e^x - \int e^x \, dx \). This simplifies to \( x e^x - e^x \).

Substitute Back and Simplify

Substitute \( \int x e^x \, dx = x e^x - e^x \) back into the equation from Step 2: \( x^2 e^x - 2(x e^x - e^x) \). Simplifying, we get \( x^2 e^x - 2x e^x + 2e^x = e^x(x^2 - 2x + 2) \).

Write the Final Result

The result of the integral is \( \int x^2 e^x \, dx = e^x (x^2 - 2x + 2) + C \), where \( C \) is the constant of integration.

Key concepts

calculus tutorial

Understanding integration by parts can be challenging, but breaking it down into manageable steps can simplify the process. Integration by parts is a technique derived from the product rule for differentiation, which we use to integrate products of functions. Think of it as reverse product rule for integration.

The fundamental formula for integration by parts is:

• \( \int u \, dv = uv - \int v \, du \)

In this method, we choose portions of the integrand (the function we're integrating) as \( u \) and \( dv \), then find their respective derivatives and integrals, \( du \) and \( v \). This sets the stage for applying the formula to simplify and resolve the integration.

The best way to solve integration by parts problems is to break them down into structured steps, usually involving repeated applications depending on the complexity of the integral. The example we're working with requires two applications to fully resolve.

integration techniques

Integration techniques are strategies used to solve integrals that are not straightforward. One of these techniques, integration by parts, is particularly helpful for integrals involving products of algebraic and exponential functions. In the original exercise, the integral \( \int x^{2} e^{x} \, dx \) is a perfect candidate for this method.

Why Choose Integration by Parts?
When you see a function multiplied by an exponential (or trigonometric function), integration by parts often simplifies your work. For example, in our exercise, the function \( x^2 \) is polynomial and \( e^x \) is exponential, making it suitable for integration by parts.

Applying Integration by Parts Multiple Times
Sometimes, the initial application of integration by parts will not completely solve the integral. In these cases, as with our example, you need to repeat the process:

• First, select suitable \( u \) and \( dv \) for the integral.
• Compute \( du \) and \( v \).
• Apply the formula to break the integral into simpler parts.
• If the remaining integral is still complex, apply integration by parts again.

Ultimately, repetition helps reduce the integral to a manageable form, leading to a solution.

step-by-step solution

Breaking down complex mathematical problems into step-by-step solutions makes them easier to tackle. Here's a simplified walkthrough of how to evaluate the integral \( \int x^2 e^x \, dx \) using the integration by parts technique twice.

Step 1: Initialization
Choose \( u = x^2 \) and \( dv = e^x \, dx \). Then compute \( du = 2x \, dx \) and \( v = e^x \). Having these components ready allows you to substitute them into the integration by parts formula.

Step 2: First Application
Substitute into the formula: \( \int x^2 e^x \, dx = x^2 e^x - \int e^x (2x) \, dx \). This reduces to \( x^2 e^x - 2 \int x e^x \, dx \).

Step 3: Second Application
For \( \int x e^x \, dx \), pick \( u = x \) and \( dv = e^x \, dx \). Then \( du = dx \) and \( v = e^x \). Substitute again to get \( x e^x - \int e^x \, dx \), which simplifies to \( x e^x - e^x \).

Step 4: Combine and Simplify
Plug the result back into the equation: \( x^2 e^x - 2(x e^x - e^x) \). Simplifying, you obtain \( e^x(x^2 - 2x + 2) \).

This comprehensive step-by-step approach provides a clear solution. Following each step ensures that no part of the problem is overlooked, making integration by parts a reliable technique for solving complex integrals efficiently.