Question

question if $f(x)=int_{2}^{4x^{5}}\frac{2}{ln(t^{2})}dt$, what is $f(x)$? (do not include \$f(x)=$\ in your answer.) provide your answer below:

Explanation:

Step1: Recall the fundamental theorem of calculus

If $F(x)=\int_{a}^{g(x)}f(t)dt$, then $F^\prime(x)=f(g(x))\cdot g^\prime(x)$. Here $a = 2$, $g(x)=4x^{5}$, and $f(t)=\frac{\ln(t^{2})}{2}$.

Step2: First, find $g^\prime(x)$

Differentiate $g(x)=4x^{5}$ with respect to $x$. Using the power - rule $(x^n)^\prime=nx^{n - 1}$, we have $g^\prime(x)=20x^{4}$.

Step3: Then, find $f(g(x))$

Substitute $t = g(x)=4x^{5}$ into $f(t)$. First, simplify $f(t)=\frac{\ln(t^{2})}{2}=\ln(t)$. Then $f(g(x))=\ln(4x^{5})$.

Step4: Calculate $F^\prime(x)$

By the fundamental theorem of calculus, $F^\prime(x)=f(g(x))\cdot g^\prime(x)$. So $F^\prime(x)=\ln(4x^{5})\cdot20x^{4}$.

Answer:

$20x^{4}\ln(4x^{5})$