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{(dF)/(dx)}{\left( \int\limits_(4)^(u)(x^2+2)dx \right)}( \stackrel{(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`