# If f(3)=13, f' is continuous, and the integral from 3 to 5 f'(x)dx=24 then the value of f(5) is ?

Feb 26, 2016

$f \left(5\right) = 37$
See explanation below.

#### Explanation:

Since $\forall x , f ' \left(x\right) = f ' \left(x\right)$, $f$ is a primitive function of $f '$.

Therefore, ${\int}_{3}^{5} f ' \left(x\right) \mathrm{dx} = f \left(x\right) {|}_{3}^{5} = f \left(5\right) - f \left(3\right) = f \left(5\right) - 13 = 24$ by the fundamental theorem of calculus (part 2) (because $f '$ is continuous and thus integrable).

Thus, $f \left(5\right) = 24 + 13 = 37$.