Question

Evaluate the integral using the properties of even and odd functions as an aid. $$ \int_{-2}^{2} x\left(x^{2}+1\right)^{3} d x $$

Short answer

The integral \(\int_{-2}^{2} x(x^{2}+1)^{3} dx \) is \(0\).

Step-by-step solution

Determine whether the function is even or odd

The given function is \( x(x^{2}+1)^{3} \). If we substitute \( -x \) for \( x \), we get \(-x(-x^{2}+1)^{3}\) which simplifies to \(-x(x^{2}+1)^{3}\). So, we can see that \( f(-x) = -f(x) \), therefore, the function \( x(x^{2}+1)^{3} \) is an odd function.

Use properties of odd functions for the integral

The property of odd functions over symmetric intervals is that their integral is 0. The interval we are integrating over, from \(-2\) to \(2\), is symmetric about the origin. Therefore, we can apply this property directly.

Final Answer

Therefore, using the property of integration of odd functions over symmetric intervals, the integral of \( x(x^{2}+1)^{3} \) from \(-2\) to \(2\) is \(0\).