Question

Calculus: Help ASAP

Answer 1

$\bf \displaystyle \int\limits_{-1}^{0}~(4x^6+2x)^3(12x^5+1)\cdot dx\\\\ -------------------------------\\\\ u=4x^6+2x\implies \cfrac{du}{dx}=24x^5+2\implies \cfrac{du}{2(12x^5+1)}=dx\\\\ -------------------------------\\\\ \displaystyle \int\limits_{-1}^{0}~u^3\underline{(12x^5+1)}\cdot\cfrac{du}{2\underline{(12x^5+1)}}\implies \cfrac{1}{2}\int\limits_{-1}^{0}~u^3\cdot du\\\\ -------------------------------$

$\bf \textit{now, we'll change the bounds, using u(x)} \\\\\\ u(-1)=4(-1)^6+2(-1)\implies u(-1)=2 \\\\\\ u(0)=4(0)^6+2()\implies u(0)=0\\\\ -------------------------------\\\\ \displaystyle \cfrac{1}{2}\int\limits_{2}^{0}~u^3\cdot du\implies \left. \cfrac{1}{2}\cdot \cfrac{u^4}{4} \right]_{2}^{0}\implies \left. \cfrac{u^4}{8} \right]_{2}^{0}\implies [0]-[2]\implies -2$