定積分・広義積分計算機

この計算機は、定積分（すなわち上下限のある積分）を、広義積分も含めて、解法手順を示しながら計算を試みます。

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 $$$\int_{t^{2}}^{t}\left( \frac{\sin{\left(x \right)}}{x} \right)dx$$$

First, calculate the corresponding indefinite integral: $$$\int{\frac{\sin{\left(x \right)}}{x} d x}=\operatorname{Si}{\left(x \right)}$$$ (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.

$$$\left(\operatorname{Si}{\left(x \right)}\right)|_{\left(x=t\right)}=\operatorname{Si}{\left(t \right)}$$$

$$$\left(\operatorname{Si}{\left(x \right)}\right)|_{\left(x=t^{2}\right)}=\operatorname{Si}{\left(t^{2} \right)}$$$

$$$\int_{t^{2}}^{t}\left( \frac{\sin{\left(x \right)}}{x} \right)dx=\left(\operatorname{Si}{\left(x \right)}\right)|_{\left(x=t\right)}-\left(\operatorname{Si}{\left(x \right)}\right)|_{\left(x=t^{2}\right)}=\operatorname{Si}{\left(t \right)} - \operatorname{Si}{\left(t^{2} \right)}$$$

Answer: $$$\int_{t^{2}}^{t}\left( \frac{\sin{\left(x \right)}}{x} \right)dx=\operatorname{Si}{\left(t \right)} - \operatorname{Si}{\left(t^{2} \right)}$$$

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定積分・広義積分計算機

定積分と広義積分を手順を追って計算する

Solution