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.

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Solution

Your input: calculate $$$\int_{o}^{t}\left( \omega t \cos{\left(2 \right)} \right)dt$$$

First, calculate the corresponding indefinite integral: $$$\int{\omega t \cos{\left(2 \right)} d t}=\frac{\omega t^{2} \cos{\left(2 \right)}}{2}$$$ (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.

$$$\left(\frac{\omega t^{2} \cos{\left(2 \right)}}{2}\right)|_{\left(t=t\right)}=\frac{\omega t^{2} \cos{\left(2 \right)}}{2}$$$

$$$\left(\frac{\omega t^{2} \cos{\left(2 \right)}}{2}\right)|_{\left(t=o\right)}=\frac{o^{2} \omega \cos{\left(2 \right)}}{2}$$$

$$$\int_{o}^{t}\left( \omega t \cos{\left(2 \right)} \right)dt=\left(\frac{\omega t^{2} \cos{\left(2 \right)}}{2}\right)|_{\left(t=t\right)}-\left(\frac{\omega t^{2} \cos{\left(2 \right)}}{2}\right)|_{\left(t=o\right)}=- \frac{o^{2} \omega \cos{\left(2 \right)}}{2} + \frac{\omega t^{2} \cos{\left(2 \right)}}{2}$$$

Answer: $$$\int_{o}^{t}\left( \omega t \cos{\left(2 \right)} \right)dt=- \frac{o^{2} \omega \cos{\left(2 \right)}}{2} + \frac{\omega t^{2} \cos{\left(2 \right)}}{2}$$$

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Solution