Question

answer the following true or false. let $f(x)=\int_{3}^{e^{5x}}f(x)dx$, then $f(x)=5e^{5x}f(e^{5x})$. true false

Explanation:

Step1: Recall Leibniz Rule

For $F(x)=\int_{a(x)}^{b(x)} f(t)dt$, $F'(x)=f(b(x))\cdot b'(x)-f(a(x))\cdot a'(x)$

Step2: Identify bounds and derivative

Here, lower bound $a(x)=3$ (constant, $a'(x)=0$), upper bound $b(x)=e^{5x}$, so $b'(x)=5e^{5x}$. Replace dummy variable $t$ in $f(t)$ to avoid confusion.

Step3: Apply the rule

$F'(x)=f(e^{5x})\cdot 5e^{5x} - f(3)\cdot 0 = 5e^{5x}f(e^{5x})$

Answer:

True