Theorem: Symmetric Integral of Even Function over One Plus Even Function to Power of Odd Function

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:$\ds \int_{-a}^a \frac {\map e x} {1 + \map t x^{\map o x} } \rd x = \int_0^a \map e x \rd x$
where:
:$a$ is a positive real number
:$\map e x$ and $\map t x$ are arbitrary even functions
:$\map o x$ is an arbitrary odd function.
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Proof:
Let the integrand be denoted by:
:$\map f x = \dfrac {\map e x} {1 + \map t x^{\map o x} }$
Using Real Function is Expressible as Sum of Even Function and Odd Function, let us express $\map f x$ as:
:$(1): \quad \map f x = \map E x + \map O x$
where:
:$\map E x$ is an even function
:$\map O x$ is an odd function
defined as:
{{begin-eqn}}
{{eqn | l = \map E x
      | r = \frac {\map f x + \map f {-x} } 2
}}
{{eqn | l = \map O x
      | r = \frac {\map f x - \map f {-x} } 2
}}
{{end-eqn}}
Then:
{{begin-eqn}}
{{eqn | o = 
      | r = \int_{-a}^a \frac {\map e x} {1 + \map t x^{\map o x} } \rd x
      | c = 
}}
{{eqn | r = \int_{-a}^a \map f x \rd x
      | c = definition of $\map f x$
}}
{{eqn | r = \int_{-a}^a \paren {\map E x + \map O x} \rd x
      | c = from $(1)$
}}
{{eqn | r = \int_{-a}^a \map E x \rd x + \int_{-a}^a \map O x \rd x
      | c = Linear Combination of Definite Integrals
}}
{{eqn | r = \int_{-a}^a \map E x \rd x
      | c = Definite Integral of Odd Function
}}
{{eqn | r = 2 \int_0^a \map E x \rd x
      | c = Definite Integral of Even Function
}}
{{eqn | r = 2 \int_0^a \paren {\frac {\map f x + \map f {-x} } 2} \rd x
      | c = definition of $\map E x$
}}
{{eqn | r = \int_0^a \paren {\map f x + \map f {-x} } \rd x
      | c = simplifying
}}
{{eqn | r = \int_0^a \paren {\frac {\map e x} {1 + \map t x^{\map o x} } + \frac {\map e {-x} } {1 + \map t {-x}^{\map o {-x} } } } \rd x
      | c = definition of $\map f x$ and $\map f {-x}$ respectively
}}
{{eqn | r = \int_0^a \paren {\frac {\map e x} {1 + \map t x^{\map o x} } + \frac {\map e x} {1 + \map t x^{-\map o x} } } \rd x
      | c = {{Defof|Even Function}} and {{Defof|Odd Function}}
}}
{{eqn | r = \int_0^a \paren {\frac {\map e x} {1 + \map t x^{\map o x} } \frac {1 + \map t x^{-\map o x} } {1 + \map t x^{-\map o x} } + \frac {\map e x} {1 + \map t x^{-\map o x} } \frac {1 + \map t x^{\map o x} } {1 + \map t x^{\map o x} } } \rd x
      | c = multiplying top and bottom by $1 + \map t x^{-\map o x}$ and $1 + \map t x^{\map o x}$
}}
{{eqn | r = \int_0^a \paren {\frac {\map e x \paren {1 + \map t x^{\map o x} } } {1 + \map t x^{-\map o x} + \map t x^{\map o x} + \map t x^{\map o x} \map t x^{-\map o x} } + \frac {\map e x \paren {1 + \map t x^{-\map o x} } } {1 + \map t x^{\map o x} + \map t x^{-\map o x} + \map t x^{-\map o x} \map t x^{\map o x} } } \rd x
      | c = multiplying out
}}
{{eqn | r = \int_0^a \paren {\frac {\map e x \paren {1 + \map t x^{\map o x} } } {1 + \map t x^{-\map o x} + \map t x^{\map o x} + 1 } + \frac {\map e x \paren {1 + \map t x^{-\map o x} } } {1 + \map t x^{\map o x} + \map t x^{-\map o x} + 1} } \rd x
      | c = simplifying
}}
{{eqn | r = \int_0^a \paren {\frac {\map e x \paren {1 + \map t x^{\map o x} } } {2 + \map t x^{-\map o x} + \map t x^{\map o x} } + \frac {\map e x \paren {1 + \map t x^{-\map o x} } } {2 + \map t x^{-\map o x} + \map t x^{\map o x} } } \rd x
      | c = simplifying and rearranging
}}
{{eqn | r = \int_0^a \paren {\frac {\map e x \paren {1 + \map t x^{\map o x} } + \map e x \paren {1 + \map t x^{-\map o x} } } {2 + \map t x^{-\map o x} + \map t x^{\map o x} } } \rd x
      | c = combining terms with a common denominator
}}
{{eqn | r = \int_0^a \paren {\frac {\map e x \paren {1 + \map t x^{\map o x} + 1 + \map t x^{-\map o x} } } {2 + \map t x^{-\map o x} + \map t x^{\map o x} } } \rd x
      | c = Multiplication Distributes over Addition
}}
{{eqn | r = \int_0^a \paren {\frac {\map e x \paren {2 + \map t x^{-\map o x} + \map t x^{\map o x} } } {2 + \map t x^{-\map o x} + \map t x^{\map o x} } } \rd x
      | c = simplification
}}
{{eqn | r = \int_0^a \map e x \rd x
      | c = simplification
}}
{{end-eqn}}
Hence the result.
{{qed}}
Category:Odd Functions
Category:Even Functions
Category:Definite Integrals
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