The anti-derivative can be found by taking an indefinite integral:
F ( x ) = ∫ ( x 3 − sin ( 2 x ) ) d x = x 4 4 − − cos ( 2 x ) 2 + C {\displaystyle F(x)=\int (x^{3}-\sin(2x))\,dx={\frac {x^{4}}{4}}-{\frac {-\cos(2x)}{2}}+C}
where C {\displaystyle C} is an arbitrary constant.
Solve for C {\displaystyle C} by setting F ( 0 ) = 1 {\displaystyle F(0)=1} :
1 = 0 4 4 + cos ( 0 ) 2 + C = 1 2 + C ⟹ C = 1 2 {\displaystyle 1={\frac {0^{4}}{4}}+{\frac {\cos(0)}{2}}+C={\frac {1}{2}}+C\implies C={\frac {1}{2}}}
As a result, we obtain the desired anti-derivative F {\displaystyle F} as the following:
F ( x ) = x 4 4 + cos ( 2 x ) 2 + 1 2 {\displaystyle \color {blue}F(x)={\frac {x^{4}}{4}}+{\frac {\cos(2x)}{2}}+{\frac {1}{2}}} .
is an arbitrary constant.
by setting 
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as the following:
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