Question

Find the derivative of ∫(x²+2) dx with upper limit 6x² and lower limit 4

Answer 1

$\bf \displaystyle \int\limits_{4}^{6x^2}(x^2+2)dx\\\\ -------------------------------\\\\ u=6x^2\implies \cfrac{du}{dx}=12x\qquad \stackrel{\textit{2nd fundamental theorem of calculus}}{f'(x)=\cfrac{dF}{dx}\cdot \cfrac{du}{dx}}\\\\ -------------------------------$

$\bf \displaystyle F(x)=\int\limits_{4}^{u}(x^2+2)dx\implies f'(x)=\stackrel{\frac{dF}{dx}}{\left( \int\limits_{4}^{u}(x^2+2)dx \right)}( \stackrel{\frac{du}{dx}}{12x} ) \\\\\\ f'(x)=(u^2+2)12x\implies f'(x)=[(6x^2)^2+2]12x \\\\\\ f'(x)=(36x^4+2)12x\implies f'(x)=432x^5+24x$