An oddity - Solution

Graphs of Odd Functions with their Integrals

y= t

y=t and its integral. t is an odd function and its integral is even.

From the image it appears that the integral is an even function because if the integral graph is reflected in the y axis it results in a copy of itself.

y=sin(t)

From the image it appears that the integral is an even function.

y=t^3

t^3 and its integral. t cubed is an odd function and its integral is even.

y=t^5

t^5 and its integral. t to the power 5 is an odd function and its integral is even.

y=sinh(t) using sinh(t)=0.5(e^t-e^(-1t))

sinh(t), using sinh(t)=0.5*(e^t-e^(-1*t)), and its integral. sinh(t) is an odd function and its integral is even

The Maclaurin series for a general function, f(t), defined around t=0 and with all its derivatives defined at t=0 is given by:

The Maclaurin series for a general function, defined around t=0 and with all its derivatives defined at t=0

If a function is odd then it will only have odd powers of t in its Maclaurin series, i.e. f(0) and f''(0) and f⁽ⁿ⁾(0) for n even, must all be 0.

Integrating above we get:

The power series for the integral of an odd function. We see that the power series for the integral consists of only a constant term and even powers of t and therefore is even

providing t₀=0. If t₀ is non-zero this will result in an additional constant term.

We see that the power series for the integral consists of only a constant term (if t₀ is non-zero) and even powers of t and therefore is even. Therefore the integral of an odd function is always even, whatever the value of t₀.

Some integrals with non-zero t₀ are shown below

sinh(t), using sinh(t)=0.5*(e^t-e^(-1*t)), and its integral. sinh(t) is an odd function and its integral is even - independent of the value of t0

y=sin(t) and its integral with initial value t0=2. sin(t) is an odd function and its integral is even, independent of the value of t0

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