Como calcular essa integral definida

Para encontrar a integral dada temos que realizar os cálculos abaixo:

\(\begin{align} & \int_{{}}^{{}}{f(x)=\int_{0}^{1}{\frac{{{x}^{2}}}{4{{x}^{2}}+1}}} \\ & \int_{0}^{1}{\frac{{{x}^{2}}}{4{{x}^{2}}+1}}=\left[ \frac{x}{4}-\frac{1}{8}{{\tan }^{-1}}2x \right]_{0}^{1} \\ & \int_{0}^{1}{\frac{{{x}^{2}}}{4{{x}^{2}}+1}}=\left[ \frac{1}{4}-\frac{1}{8}{{\tan }^{-1}}2\cdot 1 \right]-\left[ \frac{0}{4}-\frac{1}{8}{{\tan }^{-1}}2\cdot 0 \right] \\ & \int_{0}^{1}{\frac{{{x}^{2}}}{4{{x}^{2}}+1}}=\frac{1}{4}-7,92 \\ & \int_{0}^{1}{\frac{{{x}^{2}}}{4{{x}^{2}}+1}}=-7,67 \\ \end{align} \)

Portanto, o valor da integral dada será \(\boxed{ - 7,67}\).