definite integration homework question

**asilvester635** · December 11th 2012, 11:36 AM

Do i use integration by substitution???

**Plato** · December 11th 2012, 12:43 PM

Originally Posted by asilvester635

Do i use integration by substitution???

What is the derivative of $\frac{(x^3+8)^3}{9}~?$

**Prove It** · December 11th 2012, 02:42 PM

Originally Posted by asilvester635

Do i use integration by substitution???

Yes you can use a substitution. Rewrite it as $\displaystyle \begin{align*} \frac{1}{3} \int_{-2}^{4}{3x^2 \left( x^3 + 8 \right)^2\, dx} \end{align*}$ then let $\displaystyle \begin{align*} u = x^3 + 8 \implies du = 3x^2 \, dx \end{align*}$ and note that $\displaystyle \begin{align*} u(-2) = 0 \end{align*}$ and $\displaystyle \begin{align*} u(4) = 72 \end{align*}$ and the integral becomes $\displaystyle \begin{align*} \frac{1}{3} \int_0^{72}{u^2\,du} \end{align*}$ .

I'm sure you can go from here.

**asilvester635** · December 13th 2012, 09:05 PM

cool but why did you ad 1/3 at the front? is it because you added 3 by x^2?

**MarkFL** · December 13th 2012, 09:18 PM

Yes, the 3 inside the integrand has to be countered by the 1/3 in front so that the net effect is to multiply by 1.

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