Calculate the definite integral.sa

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Given the function $y=f(x)$ with antiderivative $F(x)=\left\{ \begin{array}{l} {x^2} + 5x + {C_1}{\text{ for }}x \ge 1\\ {x^2} + 4x + {C_2}{\text{ for }}x < 1 \end{array} \right.\\$
Calculate $\int\limits_0^2 {f(x)dx} $

What I've tried here is to split $\int\limits_0^2 {f(x)dx}$ into $\int\limits_0^{\frac{1}{2}} {f(x)dx}+\int\limits_{\frac{1}{2}}^2 {f(x)dx},$ and that is equals to $F(\frac{1}{2})-F(0)+F(2)-F(\frac{1}{2})=F(2)-F(0)=2^2+5\cdot2-(0^2+4\cdot0)=14$

Is my solution correct?

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1 Answer

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You should split into :-

$$\int_{0}^{1}f(x)\,dx+\int_{1}^{2}f(x)\,dx$$.

Then you have $$(\lim_{x\to 1^-}F(x))-F(0)+F(2)-F(1)=5+C_{2}-C_{2}+14+C_{1}-6-C_{1}=13$$ as the answer

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