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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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.

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Solution

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

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

$$$\left(- \frac{\cos{\left(2 \theta \right)}}{2}\right)|_{\left(\theta=\frac{\pi}{2}\right)}=\frac{1}{2}$$$

$$$\left(- \frac{\cos{\left(2 \theta \right)}}{2}\right)|_{\left(\theta=0\right)}=- \frac{1}{2}$$$

$$$\int_{0}^{\frac{\pi}{2}}\left( \sin{\left(2 \theta \right)} \right)d\theta=\left(- \frac{\cos{\left(2 \theta \right)}}{2}\right)|_{\left(\theta=\frac{\pi}{2}\right)}-\left(- \frac{\cos{\left(2 \theta \right)}}{2}\right)|_{\left(\theta=0\right)}=1$$$

Answer: $$$\int_{0}^{\frac{\pi}{2}}\left( \sin{\left(2 \theta \right)} \right)d\theta=1$$$

定積分與廣義積分計算器

逐步計算定積分與瑕積分

Solution