Mathematics - Definite Integration Question with Solution | TestHub

$\underset{0}{\overset{1}{\int }}\underbrace{4x^3}\underbrace{\frac{d^2}{d{x}^2} {\left(1-x^2\right)}^5}dx$
$={\left[4x^3\frac{d}{dx} {\left(1-x^2\right)}^5\right]}_0^1- \underset{0}{\overset{1}{\int }}12x^2\frac{d}{dx} {\left(1-x^2\right)}^5 dx$
$={\left[4x^3 \times 5{\left(1-x^2\right)}^4 \left(-2x\right)\right]}_0^1-12\left[{\left[x^2{\left(1-x^2\right)}^5\right]}_0^1- \underset{0}{\overset{1}{\int }}2x{\left(1-x^2\right)}^{5}dx\right]$
$=0-0-12\left[0-0\right]+12 \underset{0}{\overset{1}{\int }}2x{\left(1-x^2\right)}^5 dx$
$=12 \times {\left[-\frac{{\left(1-x^2\right)}^6}{6}\right]}_0^1$
$=12\left[0+\frac{1}{6}\right]=2$