$xf(x^3)+f(1-x^2)=x^3+x^2+1\\ \Leftrightarrow x^2f(x^3)+xf(1-x^2)=x^4+x^3+x\\ +)\displaystyle\int\limits^0_{-1} (x^2f(x^3)+xf(1-x^2)) \, dx=\displaystyle\int\limits^0_{-1} (x^4+x^3+x)\\ \Leftrightarrow\dfrac{1}{3}\displaystyle\int\limits^0_{-1} f(x^3)\, d(x^3)-\dfrac{1}{2}\displaystyle\int\limits^0_{-1} f(1-x^2)\, d(1-x^2)=\left(\dfrac{1}{5}x^5+\dfrac{1}{4}x^4+\dfrac{1}{2}x^2\right) \Big|^0_{-1}\\ \Leftrightarrow\dfrac{1}{3}\displaystyle\int\limits^0_{-1} fx \, dx-\dfrac{1}{2}\displaystyle\int\limits^1_{0} fx \, dx=\dfrac{-11}{20}\\ +)\displaystyle\int\limits^1_{0} (x^2f(x^3)+xf(1-x^2)) \, dx=\displaystyle\int\limits^1_{0} (x^4+x^3+x)\\ \Leftrightarrow\dfrac{1}{3}\displaystyle\int\limits^1_{0} f(x^3)\, d(x^3) - \dfrac{1}{2}\displaystyle\int\limits^1_{0} f(1-x^2)\, d(1-x^2)=\left(\dfrac{1}{5}x^5+\dfrac{1}{4}x^4+\dfrac{1}{2}x^2\right) \Big|^1_{0}\\ \Leftrightarrow\dfrac{1}{3}\displaystyle\int\limits^1_{0} fx \, dx-\dfrac{1}{2}\displaystyle\int\limits^0_{1} fx \, dx=\dfrac{19}{20}\\ \Leftrightarrow\dfrac{1}{3}\displaystyle\int\limits^1_{0} fx \, dx+\dfrac{1}{2}\displaystyle\int\limits^1_{0} fx \, dx=\dfrac{19}{20}\\ \Leftrightarrow\displaystyle\int\limits^1_{0} fx \, dx=\dfrac{57}{50}\\\Rightarrow\displaystyle\int\limits^0_{-1} fx \, dx=\dfrac{3}{50}$
