Proving the Symmetry of Definite Integrals with a u-Substitution

Understanding how to manipulate and prove properties of definite integrals is a fundamental concept in calculus. One such property is the symmetry of integrals over the interval [0, 1], specifically proving that the integral of a function f(x) over [0, 1] is equal to the integral of f(1 - x) over the same interval. This article will guide you through the process of using u-substitution to prove this symmetry.

1. Introduction to the Problem

The problem we aim to solve is to prove that if f(x) is a continuous function, then

[ int_{0}^{1} f(x) , dx int_{0}^{1} f(1 - x) , dx ]

This equality can be established using a change of variables, specifically a u-substitution.

2. The Substitution Process

The first step is to apply a u-substitution to the integral on the right-hand side of the equation. We start with the integral:

[ int_{0}^{1} f(1 - x) , dx ]

Let ( u 1 - x ). This substitution will transform the integrand and the limits of integration.

2.1. The Substitution

By substituting ( u 1 - x ), we find the differential:

[ du -dx ]

This implies:

[ dx -du ]

Switching the sign of the differential will change the limits of integration as well. When ( x 0 ), then:

[ u 1 - 0 1 ]

And when ( x 1 ), then:

[ u 1 - 1 0 ]

Thus, the integral transforms from being evaluated from ( x 0 ) to ( x 1 ) to being evaluated from ( u 1 ) to ( u 0 ).

2.2. Rewriting the Integral

Now, we rewrite the integral with the new variable ( u ):

[ int_{0}^{1} f(1 - x) , dx int_{1}^{0} f(u) cdot (-du) ]

Since the differential ( du ) is negative, we can rewrite the integral as:

[ int_{1}^{0} f(u) cdot (-du) int_{0}^{1} f(u) , du ]

The negative sign in the differential cancels out due to the reversal of the limits of integration. Thus, we have:

[ int_{0}^{1} f(1 - x) , dx int_{0}^{1} f(u) , du ]

3. Conclusion

Since ( u ) is a dummy variable, the variable of integration is purely symbolic. Therefore, the integral:

[ int_{0}^{1} f(1 - x) , dx int_{0}^{1} f(u) , du int_{0}^{1} f(x) , dx ]

Consequently, we have proven that:

[ int_{0}^{1} f(x) , dx int_{0}^{1} f(1 - x) , dx ]

This symmetry property holds for any continuous function ( f(x) ) over the interval [0, 1]. The proof is now complete.

4. Summary and Key Points

The key steps in this proof are:

Change of Variables: Using ( u 1 - x ) and ( du -dx ). Determining the Limits: Switching the limits of integration accordingly. Reversing the Integration Limits: Simplify the integral by reversing the limits and removing the negative sign from the differential.

Understanding these steps and practicing them will enhance your ability to manipulate and prove properties of definite integrals. This technique is particularly useful in more complex integral problems and in understanding the symmetries within functions.