定積分・広義積分計算機
定積分と広義積分を手順を追って計算する
この計算機は、定積分（すなわち上下限のある積分）を、広義積分も含めて、解法手順を示しながら計算を試みます。
Solution
Your input: calculate $$$\int_{1}^{3}\left( \frac{\sqrt{1 - \frac{1}{x}}}{x^{2}} \right)dx$$$
First, calculate the corresponding indefinite integral: $$$\int{\frac{\sqrt{1 - \frac{1}{x}}}{x^{2}} d x}=\frac{2 \left(\frac{x - 1}{x}\right)^{\frac{3}{2}}}{3}$$$ (for steps, see indefinite integral calculator)
According to the Fundamental Theorem of Calculus, $$$\int_a^b F(x) dx=f(b)-f(a)$$$, so just evaluate the integral at the endpoints, and that's the answer.
$$$\left(\frac{2 \left(\frac{x - 1}{x}\right)^{\frac{3}{2}}}{3}\right)|_{\left(x=3\right)}=\frac{4 \sqrt{6}}{27}$$$
$$$\left(\frac{2 \left(\frac{x - 1}{x}\right)^{\frac{3}{2}}}{3}\right)|_{\left(x=1\right)}=0$$$
$$$\int_{1}^{3}\left( \frac{\sqrt{1 - \frac{1}{x}}}{x^{2}} \right)dx=\left(\frac{2 \left(\frac{x - 1}{x}\right)^{\frac{3}{2}}}{3}\right)|_{\left(x=3\right)}-\left(\frac{2 \left(\frac{x - 1}{x}\right)^{\frac{3}{2}}}{3}\right)|_{\left(x=1\right)}=\frac{4 \sqrt{6}}{27}$$$
Answer: $$$\int_{1}^{3}\left( \frac{\sqrt{1 - \frac{1}{x}}}{x^{2}} \right)dx=\frac{4 \sqrt{6}}{27}\approx 0.362887369301212$$$