Integration+Differentation

$\begin{aligned} I & = \int_0^1 4x^3 \cdot \frac {d^2}{dx^2}\left(1-x^2\right)^5 dx & \small \color{#3D99F6} \text{By integration by parts.} \\ & = 4x^3\cdot \frac d{dx} (1-x^2)^5 - 12 \int x^2 \cdot \frac d{dx} (1-x^2)^5 dx \bigg|_0^1 \\ & = 4x^3\cdot 5(1-x^2)^4\cdot(-2x) - 12 x^2 (1-x^2)^5 + 24 {\color{#3D99F6}\int x(1-x^2)^5 dx} \bigg|_0^1 & \small \color{#3D99F6} \text{Let }x = \sin \theta \implies dx = \cos \theta \ d\theta \\ & = 0 - 0 + 24 \color{#3D99F6}\int_0^\frac \pi 2 \sin \theta \cos^{11} \theta \ d\theta & \small \color{#3D99F6} \text{Let }u = \cos \theta \implies du = - \cos \theta \ d\theta \\ & = 24 \int_0^1 u^{11} \ du \\ & = 24 \times \frac {u^{12}}{12} \bigg|_0^1 \\ & = \boxed{2} \end{aligned}$

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Previous solution

$\begin{aligned} I & = \int_0^1 4x^3 \cdot \frac {d^2}{dx^2}\left(1-x^2\right)^5 dx \\ & = \int_0^1 4x^3 \cdot \frac {d^2}{dx^2}\left(1-5x^2+10x^4-10x^6+5x^8-x^{10}\right) dx \\ & = \int_0^1 4x^3 \cdot \frac d{dx}\left(-10x+40x^3-60x^5+40x^7-10x^9\right) dx \\ & = \int_0^1 4x^3 \left(-10+120x^2-300x^4+280x^6-90x^8\right) dx \\ & = \int_0^1 \left(-40x^3+480x^5-1200x^7+1120x^9-360x^{11}\right) dx \\ & = -10x^4+80x^6-150x^8+112x^{10}-30x^{12}\ \bigg|_0^1 \\ & = -10 + 80 - 150 + 112- 30 \\ & = \boxed{2} \end{aligned}$