定积分与广义积分计算器

该计算器将尝试计算定积分（即带上下限的积分），包括广义积分，并显示求解步骤。

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If you need `oo`, type inf.

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

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定积分与广义积分计算器

逐步计算定积分和广义积分

Solution