\(F\left(x\right)=\int f\left(x\right)dx=\int\left(e^x-1\right)^3dx=\int\left(e^{3x}-3e^{2x}+3e^x-1\right)dx\)
\(=\dfrac{1}{3}e^{3x}-\dfrac{3}{2}e^{2x}+3e^x-x+C\)
Do \(F\left(0\right)=-\dfrac{1}{6}\)
\(\Rightarrow\dfrac{1}{3}.e^{3.0}-\dfrac{3}{2}.e^{2.0}+3.e^0-0+C=-\dfrac{1}{6}\)
\(\Rightarrow C=-\dfrac{1}{6}+\dfrac{3}{2}-\dfrac{1}{3}-3=-2\)
\(\Rightarrow F\left(x\right)=\dfrac{1}{3}e^{3x}-\dfrac{3}{2}e^{2x}+3e^x-x-2\)
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