Find a primitive and apply the second fundamental theorem of calculus

by RoRi

October 14, 2015

Let

$f(x) = 4x^4 - 12x.$

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

$P(x) = \frac{4}{5} x^5 - 6x^2$

is a primitive of $f$ since

$P'(x) = 4x^4 - 12x = f(x).$

Then, by the second fundamental theorem of calculus we have

$\int_a^b f(x) \, dx = P(b) - P(a) = \frac{4}{5} (b^5 - a^5) - 6(b^2 - a^2).$

Stumbling Robot