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Find a primitive and apply the second fundamental theorem of calculus

Let

    \[ f(x) = 3 \sin x + 2x^5. \]

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

    \[ P(x) = -3 \cos x + \frac{1}{3}x^6 \]

is a primitive of f since

    \[ P'(x) = 3 \sin x + 2 x^5 = f(x). \]

Then, by the second fundamental theorem of calculus we have

    \begin{align*}   \int_a^b f(x) \, dx &= P(b) - P(a) \\  &= -3 (\cos b - \cos a) + \frac{1}{3}(b^6 - a^6).  \end{align*}

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