Chọn phương án C.

Bằng máy tính cầm tay ta có $$\displaystyle\int\limits_{1}^{2}x\sqrt{x^2+3}\mathrm{\,d}x\approx3,506753$$Trong khi đó:

• \(\displaystyle\int\limits_{2}^{\sqrt{7}}t\mathrm{\,d}t=\dfrac{3}{2}=1,5\)
• \(\displaystyle\int\limits_{2}^{7}t^2\mathrm{\,d}t=\dfrac{335}{3}\approx111,66\)
• \(\displaystyle\int\limits_{2}^{\sqrt{7}}t^2\mathrm{\,d}t\approx3,506753\)
• \(\displaystyle\int\limits_{2}^{\sqrt{7}}t^3\mathrm{\,d}t=\dfrac{33}{4}=8,25\)

Vậy \(I=\displaystyle\int\limits_{1}^{2}x\sqrt{x^2+3}\mathrm{\,d}x=\displaystyle\int\limits_{2}^{\sqrt{7}}t^2\mathrm{\,d}t\).

Chọn phương án C.

Với \(t=\sqrt{x^2+3}\) ta có

• \(t^2=x^2+3\Leftrightarrow2t\mathrm{d}t=2x\mathrm{d}x\Leftrightarrow t\mathrm{d}t=x\mathrm{d}x\)
• \(x=1\Rightarrow t=2\)
• \(x=2\Rightarrow t=\sqrt{7}\)

Vậy \(I=\displaystyle\int\limits_{2}^{\sqrt{7}}t\cdot t\mathrm{\,d}t=\displaystyle\int\limits_{2}^{\sqrt{7}}t^2\mathrm{\,d}t\).