Question

question 8 · 1 point given f(x) below, find f(x). f(x)=int_{ - 3}^{3x^{2}}\frac{t^{2}}{t + 7}dt provide your answer below: f(x)=□

Explanation:

Step1: Apply the fundamental theorem of calculus and chain - rule

If $F(x)=\int_{a}^{u(x)}f(t)dt$, then $F^{\prime}(x)=f(u(x))\cdot u^{\prime}(x)$. Here $a = - 3$, $u(x)=3x^{2}$, and $f(t)=\frac{t^{2}}{t + 7}$.

Step2: Find $u^{\prime}(x)$

Differentiate $u(x)=3x^{2}$ with respect to $x$. Using the power - rule, $u^{\prime}(x)=\frac{d}{dx}(3x^{2}) = 6x$.

Step3: Find $f(u(x))$

Substitute $t = u(x)=3x^{2}$ into $f(t)$. So $f(u(x))=\frac{(3x^{2})^{2}}{3x^{2}+7}=\frac{9x^{4}}{3x^{2}+7}$.

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

By the formula $F^{\prime}(x)=f(u(x))\cdot u^{\prime}(x)$, we have $F^{\prime}(x)=\frac{9x^{4}}{3x^{2}+7}\cdot6x=\frac{54x^{5}}{3x^{2}+7}$.

Answer:

$\frac{54x^{5}}{3x^{2}+7}$