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

Theorem

$\ds \int_{-a}^a \frac {\map e x} {1 + \map t x^{\map o x} } \rd x = \int_0^a \map e x \rd x$

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.

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:

\(\ds \map E x\)

\(=\)

\(\ds \frac {\map f x + \map f {-x} } 2\)

\(\ds \map O x\)

\(=\)

\(\ds \frac {\map f x - \map f {-x} } 2\)

Then:

\(\ds \)

\(\)

\(\ds \int_{-a}^a \frac {\map e x} {1 + \map t x^{\map o x} } \rd x\)

\(\ds \)

\(=\)

\(\ds \int_{-a}^a \map f x \rd x\)

definition of $\map f x$

\(\ds \)

\(=\)

\(\ds \int_{-a}^a \paren {\map E x + \map O x} \rd x\)

from $(1)$

\(\ds \)

\(=\)

\(\ds \int_{-a}^a \map E x \rd x + \int_{-a}^a \map O x \rd x\)

Linear Combination of Definite Integrals

\(\ds \)

\(=\)

\(\ds \int_{-a}^a \map E x \rd x\)

Definite Integral of Odd Function

\(\ds \)

\(=\)

\(\ds 2 \int_0^a \map E x \rd x\)

Definite Integral of Even Function

\(\ds \)

\(=\)

\(\ds 2 \int_0^a \paren {\frac {\map f x + \map f {-x} } 2} \rd x\)

definition of $\map E x$

\(\ds \)

\(=\)

\(\ds \int_0^a \paren {\map f x + \map f {-x} } \rd x\)

simplifying

\(\ds \)

\(=\)

\(\ds \int_0^a \paren {\frac {\map e x} {1 + \map t x^{\map o x} } + \frac {\map e {-x} } {1 + \map t {-x}^{\map o {-x} } } } \rd x\)

definition of $\map f x$ and $\map f {-x}$ respectively

\(\ds \)

\(=\)

\(\ds \int_0^a \paren {\frac {\map e x} {1 + \map t x^{\map o x} } + \frac {\map e x} {1 + \map t x^{-\map o x} } } \rd x\)

Definition of Even Function and Definition of Odd Function

\(\ds \)

\(=\)

\(\ds \int_0^a \paren {\frac {\map e x} {1 + \map t x^{\map o x} } \frac {1 + \map t x^{-\map o x} } {1 + \map t x^{-\map o x} } + \frac {\map e x} {1 + \map t x^{-\map o x} } \frac {1 + \map t x^{\map o x} } {1 + \map t x^{\map o x} } } \rd x\)

multiplying top and bottom by $1 + \map t x^{-\map o x}$ and $1 + \map t x^{\map o x}$

\(\ds \)

\(=\)

\(\ds \int_0^a \paren {\frac {\map e x \paren {1 + \map t x^{\map o x} } } {1 + \map t x^{-\map o x} + \map t x^{\map o x} + \map t x^{\map o x} \map t x^{-\map o x} } + \frac {\map e x \paren {1 + \map t x^{-\map o x} } } {1 + \map t x^{\map o x} + \map t x^{-\map o x} + \map t x^{-\map o x} \map t x^{\map o x} } } \rd x\)

multiplying out

\(\ds \)

\(=\)

\(\ds \int_0^a \paren {\frac {\map e x \paren {1 + \map t x^{\map o x} } } {1 + \map t x^{-\map o x} + \map t x^{\map o x} + 1 } + \frac {\map e x \paren {1 + \map t x^{-\map o x} } } {1 + \map t x^{\map o x} + \map t x^{-\map o x} + 1} } \rd x\)

simplifying

\(\ds \)

\(=\)

\(\ds \int_0^a \paren {\frac {\map e x \paren {1 + \map t x^{\map o x} } } {2 + \map t x^{-\map o x} + \map t x^{\map o x} } + \frac {\map e x \paren {1 + \map t x^{-\map o x} } } {2 + \map t x^{-\map o x} + \map t x^{\map o x} } } \rd x\)

simplifying and rearranging

\(\ds \)

\(=\)

\(\ds \int_0^a \paren {\frac {\map e x \paren {1 + \map t x^{\map o x} } + \map e x \paren {1 + \map t x^{-\map o x} } } {2 + \map t x^{-\map o x} + \map t x^{\map o x} } } \rd x\)

combining terms with a common denominator

\(\ds \)

\(=\)

\(\ds \int_0^a \paren {\frac {\map e x \paren {1 + \map t x^{\map o x} + 1 + \map t x^{-\map o x} } } {2 + \map t x^{-\map o x} + \map t x^{\map o x} } } \rd x\)

Multiplication Distributes over Addition

\(\ds \)

\(=\)

\(\ds \int_0^a \paren {\frac {\map e x \paren {2 + \map t x^{-\map o x} + \map t x^{\map o x} } } {2 + \map t x^{-\map o x} + \map t x^{\map o x} } } \rd x\)

simplification

\(\ds \)

\(=\)

\(\ds \int_0^a \map e x \rd x\)

simplification

Hence the result.

$\blacksquare$