Question

Evaluate each integral. $$\int x^{2} e^{-x} d x$$

   Evaluate each integral.
$$\int x^{2} e^{-x} d x$$

Higher Level Mathematics

Higher Level Mathematics

Ibrahim Wazir, Tim… 2012 Edition

Chapter 16, Problem 2

Instant Answer

Solved by Expert Varsha Aggarwal

12/16/2021

Step 1

Here, we can let $u = x^{2}$ and $dv = e^{-x} dx$. Show more…

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Evaluate each integral. $$\int x^{2} e^{-x} d x$$

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Key Concepts

Integration by Parts

Integration by parts is a fundamental technique derived from the product rule of differentiation. It is used to integrate products of two functions by choosing one function to differentiate (reducing its complexity) and another to integrate. This method is particularly useful when one of the functions is a polynomial and the other is an exponential or trigonometric function.

Reduction Formula

Reduction formulas allow you to simplify integrals by reducing the power of a function (often a polynomial) step-by-step. In integrals involving polynomials multiplied by exponential functions, a reduction formula can recursively decrease the polynomial degree, converting a more complex integral into simpler ones that are easier to evaluate.

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