Answer
$0.1876$
Work Step by Step
$sin(x^{4})=\Sigma_{n=0}^{\infty}\dfrac{(-1)^{n}x^{8n+4}}{(2n+1)!}$
$\int_{0}^{1}sin(x^{4})dx=\int_{0}^{1}\Sigma_{n=0}^{\infty}\dfrac{(-1)^{n}x^{8n+4}}{(2n+1)!}$
Then
$=\Sigma_{n=0}^{\infty}\dfrac{(-1)^{n}x^{8n+5}}{(8n+5)(2n+1)!}]_{0}^{1}$
$=[\frac{x^{5}}{5}-\frac{x^{13}}{13(3!)}+\frac{x^{21}}{21(5!)}]_{0}^{1}$
$\approx 0.1876$