Question

Evaluate the integral. $\int_0^1\left(x^2+1\right) e^{-x} d x$

   Evaluate the integral.
$\int_0^1\left(x^2+1\right) e^{-x} d x$

Calculus

James Stewart 7th Edition

Chapter 7, Problem 24

Instant Answer

Solved by Expert Wen Zheng

01/31/2018

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Step 1: We need to evaluate the integral $\int_0^1(x^2+1)e^{-x}dx$. Show more…

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Evaluate the integral. $\int_0^1\left(x^2+1\right) e^{-x} d x$

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Key Concepts

Exponential Function Integration

Exponential functions exhibit unique properties that make them straightforward to differentiate and integrate. When an exponential function is multiplied by a polynomial, as is common in many integrals, using methods like integration by parts systematically breaks down the integral into more manageable pieces.

Definite Integral

A definite integral represents the accumulation of quantities, such as area under a curve, over a specified interval. Its evaluation requires finding an antiderivative (primitive function) and then applying the Fundamental Theorem of Calculus by subtracting the value of the antiderivative at the lower limit from that at the upper limit.

Integration by Parts

Integration by parts is a technique derived from the product rule of differentiation. It allows the integration of products of functions by transferring the derivative from one function to another, thereby reducing the complexity of the integral into simpler parts.

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