牛顿-莱布尼茨公式求积分例题

1

求

$$\int_{0}^{1}\frac{1}{1+{x}^{2}}dx$$

已知$\frac{\mathrm{d} }{\mathrm{d} x}arctanx=\frac{1}{1+{x}^{2}}$
所以结果$=arctan1-arctan0=\frac{\pi }{4}$

注：如何求$\frac{\mathrm{d} }{\mathrm{d} x}arctanx$

$$y=arctanx;$$
$$\therefore x=tany;$$
$$\therefore 1= \frac{\mathrm{d} }{\mathrm{d} x}tany=tan^{'}y.y^{'};$$
$$\therefore y^{'}=1/tan^{'}y$$

现在求

$$\left (tany \right )^{'}$$

$$\left (tany \right )^{'}=\left (\frac{siny}{cosy} \right )^{'}= \frac{sin^{'}ycosy-sinycos^{'}y}{cos^{2}y} =\frac{sin^{2}y+cos^{2}y}{cos^{2}y}$$
$$y^{'}=\frac{1}{tan^{'}y}=\frac{cos^{2}y}{sin^{2}y+cos^{2}y}=\frac{1}{(tan^2y+1)}=\frac{1}{1+x^2}$$