Symmetry in integrals Use symmetry to evaluate the following i...

Question

Symmetry in integrals Use symmetry to evaluate the following integrals.

∫²₋₂ [(x³ ― 4x) / (x² + 1)] dx

Accepted Answer

Welcome back, everyone. Evaluate the integral from -3 to 3 of 2 X to the power of 5 minus 6 X cubed, all divided by X to the power of 4 + 1 multiplied by DX using symmetry. For this problem, we have to focus on our function that we're integrating. Let's suppose that F of X is equal to 2 X to the power of 5 minus 6 X cubed divided by X to the power of 4 + 1. Let's consider any possible symmetries. For that purpose, we want to evaluate F of negative X to classify this function as either even or odd, right, if it contains symmetry. So we get 2 multiplied by negative X to the power of 5 minus 6 multiplied by negative X to the power of 3. Divided by negative X to the power of 4 + 1. Now if we have an odd exponent, we can essentially state that negative X raised to the power of 5 can be written as X to the power of 5 with a negative sign in front, right, because if we raise a negative number to an odd exponent, we get a negative number. So this is equal to -2 X to the power of 5 minus minus means plus 6 X cubed divided by. Now in the denominator we have an even exponent that's 4. So when we raise a negative number to the power of 4, we get a positive number. So we can simply state X to the power of 4 + 1. That's our denominator. And now let's notice that we can factor out the negative sign, right? We can say that this is negative of 2x the power of 5 minus 6 X cubed. Divided by x to the power of 4 + 1. Notice that we have a negative sign in front of the original F of X. This is negative of F of X. And the conclusion is that this function is odd. Now whenever we're integrating an odd function from. Negative constant up to the same positive constant such as -3 up to 3. It can be any other number, but we want to ensure that we have two opposite numbers right at the lower bound and the upper bound. The integral of any odd function for this condition would be zero, so we can say that the integral from -3 up to 3 of 2 X to the power of 5 minus. 6 x cubed all divided by x to the power of 4 + 1. The X is equal to 0 because our function is odd. Thank you for watching.

Symmetry in integrals Use symmetry to evaluate the following integrals.
∫²₋₂ [(x³ ― 4x) / (x² + 1)] dx

Key Concepts

Symmetry in Functions

Definite Integrals

Properties of Integrals

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