Q. 8 - Calculus

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 $x\in [a, b]$,

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

Step-by-Step Solution

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 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 $x \in [a, b]$,  

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