# What is the integration of I = int(1-2x³).x².dx by using the correct substitution for t = _ ?

Jun 12, 2018

$= - \frac{1}{12} {\left(1 - 2 {x}^{3}\right)}^{2} + C$

#### Explanation:

$\int \left(1 - 2 {x}^{3}\right) {x}^{2} \mathrm{dx}$

Use the substitution: $t = 1 - 2 {x}^{3}$.

It follows that: $\mathrm{dt} = - 6 {x}^{2} \mathrm{dx} \implies - \frac{1}{6} \mathrm{dt} = {x}^{2} \mathrm{dx}$

Substituting this in gives us:

$\int \left(1 - 2 {x}^{3}\right) {x}^{2} \mathrm{dx} = \int - \frac{1}{6} t \mathrm{dt}$

$= - \frac{1}{12} {t}^{2} + C$

Reverse the substitution to get:

$= - \frac{1}{12} {\left(1 - 2 {x}^{3}\right)}^{2} + C$