Answer:
-162
Explanation:
Integration by substitution is a technique in calculus where a change of variables is made to simplify the integral by expressing it in terms of a new variable, making the integration easier to manage.
To evaluate the following integral, use the method of substitution:

Let u = 2x⁴ + 8x.
Differentiate u with respect to x using the power rule for differentiation by multiplying each term by its exponent and then subtracting 1 from the exponent:


Rewrite it so that dx is on its own:


Change the limits of integration from x to u:


Rewrite the integral in terms of u:

Integrate with respect to u using the power rule for integration by adding 1 to the exponent of each term and then dividing by the new exponent:
![\begin{aligned}\displaystyle \int_(-6)^0(u^3)/(2) &=\left[(u^(3+1))/(2 \cdot (3+1))\right]^0_(-6)\\\\&=\left[(u^(4))/(8)\right]^0_(-6)\\\\&=((0)^4)/(8)-((-6)^4)/(8)\\\\&=0-(1296)/(8)\\\\&=-162\end{aligned}](https://img.qammunity.org/2018/formulas/mathematics/high-school/8gqpbqqsnua4s8gvo2ef2bmpy6czu7ec58.png)
Therefore, the value of the given integral is -162.