# How do you calculate the derivative of intcos(t)/t dt from [x,3]?

Jun 30, 2015

#### Explanation:

Fundamental Theorem of Calculus Part 1 says (in part) that if $f$ is continuous on $\left[a , b\right]$ and function $g$ is defined on $\left[a , b\right]$ by

$g \left(x\right) = {\int}_{a}^{x} f \left(t\right) \mathrm{dt}$, then $g ' \left(x\right) = f \left(x\right)$ (for all $x \in \left(a , b\right)$)

$g \left(x\right) = {\int}_{x}^{3} \cos \frac{t}{t} \mathrm{dt}$

Clearly, in this question we have $f \left(t\right) = \cot \frac{t}{t}$.

Notice that FTC 1 requires the constant to be the lower limit of integration, so we use the properties of definite integral to write:

$g \left(x\right) = - {\int}_{3}^{x} \cos \frac{t}{t} \mathrm{dt}$

Now, we can see that $g ' \left(x\right) = - \cos \frac{x}{x}$.

That's it. Stop! We are finished. Move on to the next question.
(On exams, the trickiest thing about this kind of question is often understanding how easy it is. We expect more work to be needed.)

example
Find the derivative of: $g \left(x\right) = {\int}_{1}^{x} \sin \frac{\frac{2 \pi t}{3}}{\sqrt{{t}^{3} - 2 t}} \mathrm{dt}$

It is: $g ' \left(x\right) = \sin \frac{\frac{2 \pi x}{3}}{\sqrt{{x}^{3} - 2 x}}$.

(It is so easy, that graders may penalize heavily for forgetting to change the variable $t$ to $x$.)