Tìm nguyên hàm $\Large F(x)=\int (x+sin x)\mathrm{d}x$ biết $\Large F(

Tìm nguyên hàm $\Large F(x)=\int (x+sin x)\mathrm{d}x$ biết $\Large F(0)=1.$

• A.

  $\Large F(x)=x^2-cosx+20.$

• B.

  $\Large F(x)=\dfrac{1}{2}x^2-cosx.$

• C.

  $\Large F(x)=\dfrac{1}{2}x^2-cosx+2.$

• D.

  $\Large F(x)=x^2+cosx+20.$

Chọn C

Ta có $\Large F(x)=\int (x+sinx)\mathrm{d}x=\int x\mathrm{d}x+\int sinx\mathrm{d}x=\dfrac{x^2}{2}-cosx+C.$

Mặt khác ta có $\Large F(0)=1 \Leftrightarrow \dfrac{0^2}{2}-cos0+C=1 \Leftrightarrow C=2.$

Vậy $\Large F(x)=\dfrac{x^2}{2}-cosx+2.$