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Chapter 4: Problem 78

Analyzing the Integrand Without integrating, explain why $$\int_{-2}^{2} x\left(x^{2}+1\right)^{2} d x=0$$

Short Answer

Expert verified
The integral \(\int_{-2}^{2} x(x^{2}+1)^{2} dx\) is 0 because the function being integrated is odd over the symmetric interval from -2 to 2.

Step by step solution

01

Identify the type of function

Identify the function that is to be integrated, \(x(x^2+1)^2\), as an odd or even function. This function is odd because substituting -x for x gives \(-x((-x)^2+1)^2 = -x(x^2+1)^2\), which is the negative of the original function.
02

Reasoning with properties of odd function

Recognize that the integral of an odd function over a symmetric interval is 0. This is because the negative region from -a to 0 will exactly cancel out the positive region from 0 to a.
03

Apply the property

Apply the property of integrals and odd functions that was recognized in the previous step. The integral \(\int_{-2}^{2} x(x^{2}+1)^{2} dx\) is equal to 0, because it represents the integral of an odd function over a symmetric interval around 0.

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