Let f(x) be a continuous function defined on [a, b], then a function φ(x) defined by for all x ϵ [a, b] is called the integral function of the function f(x).
Property I: The integral function of an integrable function is always continuous.
Property II: if φ(x) is differentiable on (a, b) and φ’(x) = f(x) for all x ϵ (a, b).
Property III: The integral function of an odd function is an even function.
If f(x) is an odd function, then is an function.
Example: Find the greatest value of in the interval [5π/3, 7π/4]
Solution: We have,
⇒ F’(x) = 6 cosx - 2sinx
For all x ϵ [5π/3, 7π/4], we have
⇒ cos x > 0 and sin x < 0
⇒ F’(x) = 6 cosx - 2sinx > 0
⇒ F(x) is an increasing function on [5π/3,
7π/4]
⇒ F(x) attains greatest value at x = 7π/4.
Hence,
Greatest value = F (7π/4)
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