定積分與廣義積分計算器

此計算器將嘗試計算定積分（即帶有上下限的積分），包含廣義積分，並顯示步驟。

Enter a function:

Integrate with respect to:

Enter a lower limit:

If you need `-oo`, type -inf.

Enter an upper limit:

If you need `oo`, type inf.

Please write without any differentials such as `dx`, `dy` etc.

If the calculator did not compute something or you have identified an error, or you have a suggestion/feedback, please contact us.

Solution

Your input: calculate $$$\int_{3}^{9}\left( \frac{\sqrt{y}}{2} + \frac{1}{2 \sqrt{y}} \right)dy$$$

First, calculate the corresponding indefinite integral: $$$\int{\left(\frac{\sqrt{y}}{2} + \frac{1}{2 \sqrt{y}}\right)d y}=\frac{\sqrt{y} \left(y + 3\right)}{3}$$$ (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.

$$$\left(\frac{\sqrt{y} \left(y + 3\right)}{3}\right)|_{\left(y=9\right)}=12$$$

$$$\left(\frac{\sqrt{y} \left(y + 3\right)}{3}\right)|_{\left(y=3\right)}=2 \sqrt{3}$$$

$$$\int_{3}^{9}\left( \frac{\sqrt{y}}{2} + \frac{1}{2 \sqrt{y}} \right)dy=\left(\frac{\sqrt{y} \left(y + 3\right)}{3}\right)|_{\left(y=9\right)}-\left(\frac{\sqrt{y} \left(y + 3\right)}{3}\right)|_{\left(y=3\right)}=12 - 2 \sqrt{3}$$$

Answer: $$$\int_{3}^{9}\left( \frac{\sqrt{y}}{2} + \frac{1}{2 \sqrt{y}} \right)dy=12 - 2 \sqrt{3}\approx 8.53589838486224$$$

Please try a new game Rotatly

定積分與廣義積分計算器

逐步計算定積分與瑕積分

Solution