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 $$$\int_{x}^{0}\left( \frac{1}{a - b \sqrt{x}} \right)dx$$$
First, calculate the corresponding indefinite integral: $$$\int{\frac{1}{a - b \sqrt{x}} d x}=\frac{2 \left(- a \ln{\left(\left|{a - b \sqrt{x}}\right| \right)} - b \sqrt{x}\right)}{b^{2}}$$$ (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.
$$$\left(\frac{2 \left(- a \ln{\left(\left|{a - b \sqrt{x}}\right| \right)} - b \sqrt{x}\right)}{b^{2}}\right)|_{\left(x=0\right)}=- \frac{2 a \ln{\left(\left|{a}\right| \right)}}{b^{2}}$$$
$$$\left(\frac{2 \left(- a \ln{\left(\left|{a - b \sqrt{x}}\right| \right)} - b \sqrt{x}\right)}{b^{2}}\right)|_{\left(x=x\right)}=\frac{2 \left(- a \ln{\left(\left|{a - b \sqrt{x}}\right| \right)} - b \sqrt{x}\right)}{b^{2}}$$$
$$$\int_{x}^{0}\left( \frac{1}{a - b \sqrt{x}} \right)dx=\left(\frac{2 \left(- a \ln{\left(\left|{a - b \sqrt{x}}\right| \right)} - b \sqrt{x}\right)}{b^{2}}\right)|_{\left(x=0\right)}-\left(\frac{2 \left(- a \ln{\left(\left|{a - b \sqrt{x}}\right| \right)} - b \sqrt{x}\right)}{b^{2}}\right)|_{\left(x=x\right)}=- \frac{2 a \ln{\left(\left|{a}\right| \right)}}{b^{2}} - \frac{2 \left(- a \ln{\left(\left|{a - b \sqrt{x}}\right| \right)} - b \sqrt{x}\right)}{b^{2}}$$$
Answer: $$$\int_{x}^{0}\left( \frac{1}{a - b \sqrt{x}} \right)dx=- \frac{2 a \ln{\left(\left|{a}\right| \right)}}{b^{2}} - \frac{2 \left(- a \ln{\left(\left|{a - b \sqrt{x}}\right| \right)} - b \sqrt{x}\right)}{b^{2}}$$$