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Find a primitive and apply the second fundamental theorem of calculus

by RoRi

Let

    \[ f(x) = x^{\frac{4}{3}} - 5 \cos x. \]

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

    \[ P(x) = \frac{3}{7} x^{\frac{7}{3}} - 5 \sin x \]

is a primitive of f since

    \[ P'(x) = x^{\frac{4}{3}} - 5 \cos x = f(x). \]

Then, by the second fundamental theorem of calculus we have

    \begin{align*} \int_a^b f(x) \, dx &= P(b) - P(a) \\ &= \frac{3}{7} \left( b^{\frac{7}{3}} - a^{\frac{7}{3}} \right) - 5 (\sin b - \sin a). \end{align*}

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