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Find a primitive and apply the second fundamental theorem of calculus

Let

    \[ f(x) = 5x^3. \]

Find a function P(x) such that P'(x) = f(x) (i.e., a primitive of f). Use the second fundamental theorem of calculus to evaluate

    \[ \int_a^b f(x) \, dx. \]


The function

    \[ P(x) = \frac{5}{4} x^4 \]

is a primitive of f since

    \[ P'(x) = 5x^3 = f(x). \]

Then, by the second fundamental theorem of calculus we have

    \[ \int_a^b f(x) \, dx = P(b) - P(a) = \frac{5}{4} (b^4 - a^4). \]

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