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。

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

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If you need `oo`, type inf.

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Solution

Your input: calculate $$$\int_{0}^{\pi}\left( \frac{x \sin{\left(x \right)}}{2} \right)dx$$$

First, calculate the corresponding indefinite integral: $$$\int{\frac{x \sin{\left(x \right)}}{2} d x}=- \frac{x \cos{\left(x \right)}}{2} + \frac{\sin{\left(x \right)}}{2}$$$ (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.

$$$\left(- \frac{x \cos{\left(x \right)}}{2} + \frac{\sin{\left(x \right)}}{2}\right)|_{\left(x=\pi\right)}=\frac{\pi}{2}$$$

$$$\left(- \frac{x \cos{\left(x \right)}}{2} + \frac{\sin{\left(x \right)}}{2}\right)|_{\left(x=0\right)}=0$$$

$$$\int_{0}^{\pi}\left( \frac{x \sin{\left(x \right)}}{2} \right)dx=\left(- \frac{x \cos{\left(x \right)}}{2} + \frac{\sin{\left(x \right)}}{2}\right)|_{\left(x=\pi\right)}-\left(- \frac{x \cos{\left(x \right)}}{2} + \frac{\sin{\left(x \right)}}{2}\right)|_{\left(x=0\right)}=\frac{\pi}{2}$$$

Answer: $$$\int_{0}^{\pi}\left( \frac{x \sin{\left(x \right)}}{2} \right)dx=\frac{\pi}{2}\approx 1.5707963267949$$$

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Solution