Home » Blog » Find a primitive and apply the second fundamental theorem of calculus

Find a primitive and apply the second fundamental theorem of calculus

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*}

Point out an error, ask a question, offer an alternative solution (to use Latex type [latexpage] at the top of your comment):