WAEC 2022 · Paper 1 · Q10

Evaluate: $\displaystyle\int_0^1 x^2\left(x^3 + 2\right)^3 dx$ .

Worked solution (try it first)

The $x^2$ is a multiple of the derivative of $x^3 + 2$ , so let $u = x^3 + 2$ .
Then $du = 3x^2\,dx$ and $x^2\,dx = \frac13\,du$ .
$\displaystyle\int x^2(x^3 + 2)^3\,dx = \frac13 \times \frac{u^4}{4}$
$= \frac{(x^3 + 2)^4}{12}$ .
Between the limits: $\dfrac{3^4 - 2^4}{12} = \dfrac{81 - 16}{12}$
$= \dfrac{65}{12}$ , option B.