Math Problem Statement

Evaluate the integral: ∫(x^2 e^x) / (x + 2)^2 dx

Solution

The integral in the image is:

x2ex(x+2)2dx\int \frac{x^2 e^x}{(x+2)^2} \, dx

Steps to Solve:

This integral involves a combination of exponential and rational functions. It may be solved using substitution methods or by simplifying through algebraic manipulation.

Let me solve this for you step by step.


  1. Substitution:

    Let u=x+2u = x + 2. Then: du=dxandx=u2du = dx \quad \text{and} \quad x = u - 2

    Substituting these into the integral:

    (u2)2eu2u2du\int \frac{(u-2)^2 e^{u-2}}{u^2} \, du

  2. Expand and Simplify:

    Rewrite (u2)2eu2(u-2)^2 e^{u-2} as: (u2)2eu2=(u24u+4)eu2(u-2)^2 e^{u-2} = (u^2 - 4u + 4) e^{u-2}

    Factor out eu2e^{u-2} to simplify computation.


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Math Problem Analysis

Mathematical Concepts

Calculus
Integration
Substitution Method
Exponential Functions

Formulas

u-substitution: u = x + 2, du = dx

Theorems

Fundamental Theorem of Calculus

Suitable Grade Level

Grades 11-12, College Level