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。

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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 $$$\int_{0}^{\frac{\pi}{4}}\left( \cot{\left(x + \frac{\pi}{4} \right)} \right)dx$$$

First, calculate the corresponding indefinite integral: $$$\int{\cot{\left(x + \frac{\pi}{4} \right)} d x}=\ln{\left(\left|{\sin{\left(x + \frac{\pi}{4} \right)}}\right| \right)}$$$ (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.

$$$\left(\ln{\left(\left|{\sin{\left(x + \frac{\pi}{4} \right)}}\right| \right)}\right)|_{\left(x=\frac{\pi}{4}\right)}=0$$$

$$$\left(\ln{\left(\left|{\sin{\left(x + \frac{\pi}{4} \right)}}\right| \right)}\right)|_{\left(x=0\right)}=\ln{\left(\frac{\sqrt{2}}{2} \right)}$$$

$$$\int_{0}^{\frac{\pi}{4}}\left( \cot{\left(x + \frac{\pi}{4} \right)} \right)dx=\left(\ln{\left(\left|{\sin{\left(x + \frac{\pi}{4} \right)}}\right| \right)}\right)|_{\left(x=\frac{\pi}{4}\right)}-\left(\ln{\left(\left|{\sin{\left(x + \frac{\pi}{4} \right)}}\right| \right)}\right)|_{\left(x=0\right)}=- \ln{\left(\frac{\sqrt{2}}{2} \right)}$$$

Answer: $$$\int_{0}^{\frac{\pi}{4}}\left( \cot{\left(x + \frac{\pi}{4} \right)} \right)dx=- \ln{\left(\frac{\sqrt{2}}{2} \right)}\approx 0.346573590279973$$$

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