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 $$$\int_{0}^{1}\left( t^{2} x - t^{2} y \right)dx$$$

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

$$$\left(\frac{t^{2} x \left(x - 2 y\right)}{2}\right)|_{\left(x=1\right)}=\frac{t^{2} \left(1 - 2 y\right)}{2}$$$

$$$\left(\frac{t^{2} x \left(x - 2 y\right)}{2}\right)|_{\left(x=0\right)}=0$$$

$$$\int_{0}^{1}\left( t^{2} x - t^{2} y \right)dx=\left(\frac{t^{2} x \left(x - 2 y\right)}{2}\right)|_{\left(x=1\right)}-\left(\frac{t^{2} x \left(x - 2 y\right)}{2}\right)|_{\left(x=0\right)}=\frac{t^{2} \left(1 - 2 y\right)}{2}$$$

Answer: $$$\int_{0}^{1}\left( t^{2} x - t^{2} y \right)dx=\frac{t^{2} \left(1 - 2 y\right)}{2}$$$

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Solution