Calculus

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Let f(t) be an odd function

Using properties of the definite integral plus simple
substitution show that if f is continuous on [−a, a] for a positive number a, then
Z a
−a
f(t) dt = 0

  • Calculus -

    The definition of an odd function is
    f(-x)=-f(x).

    If f(x) is continuous on the interval [-a,a], then
    I=∫f(t)dt from -a to a
    = I1 + I2
    where
    I2 = ∫f(t)dt from 0 to a
    and
    I1 = ∫f(t)dt from -a to 0
    Using property of odd function, f(-x)=-f(-x), and substitute u=-t,dt=-du
    we get
    I1 = ∫-f(u)(-du) from a to 0
    =-∫f(u)du from 0 to a
    =-I2
    Thus
    I=I1+I2=-I2+I2=0


    Incidentally, as exercise, you can use the above fact to solve:
    ∫[x^3 + x^4(tan^3(x))]dx from -π/4 to π/4.

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