Question

Fill in the blanks to complete each of the following theorem statements:

8. If $f$is on $[a,b]$and $u\left(x\right)$is on $[a,b]$, then for all $x\in [a,b]$,

$\frac{d}{dx}{\int }_a^{u\left(x\right)}f\left(t\right)dt=\_\_\_\_$

Short answer

If $f$is on $[a,b]$and $u\left(x\right)$is on$[a,b]$, then for all$x\in [a,b]$

$\frac{d}{dx}{\int }_a^{u\left(x\right)}f\left(t\right)dt=f\left(u\right(x\left)\right)u\text{'}\left(x\right)$

Step-by-step solution

Step 1. Given information

We have to complete the statement,

If $f$is on $[a,b]$and $u\left(x\right)$is on $[a,b]$, then for all$x\in [a,b]$,

$\frac{d}{dx}{\int }_a^{u\left(x\right)}f\left(t\right)dt=\_\_\_\_$

Step 2. Fill in the blanks

If $f$is on $[a,b]$and $u\left(x\right)$is on $[a,b]$, then for all$x\in [a,b]$,

$\frac{d}{dx}{\int }_a^{u\left(x\right)}f\left(t\right)dt=f\left(u\right(x\left)\right)u\text{'}\left(x\right)$