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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Solution

Your input: calculate $$$\int_{\frac{\pi}{3}}^{\frac{\pi}{2}}\left( 36 \cos^{2}{\left(\theta \right)} \right)d\theta$$$

First, calculate the corresponding indefinite integral: $$$\int{36 \cos^{2}{\left(\theta \right)} d \theta}=18 \theta + 9 \sin{\left(2 \theta \right)}$$$ (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.

$$$\left(18 \theta + 9 \sin{\left(2 \theta \right)}\right)|_{\left(\theta=\frac{\pi}{2}\right)}=9 \pi$$$

$$$\left(18 \theta + 9 \sin{\left(2 \theta \right)}\right)|_{\left(\theta=\frac{\pi}{3}\right)}=\frac{9 \sqrt{3}}{2} + 6 \pi$$$

$$$\int_{\frac{\pi}{3}}^{\frac{\pi}{2}}\left( 36 \cos^{2}{\left(\theta \right)} \right)d\theta=\left(18 \theta + 9 \sin{\left(2 \theta \right)}\right)|_{\left(\theta=\frac{\pi}{2}\right)}-\left(18 \theta + 9 \sin{\left(2 \theta \right)}\right)|_{\left(\theta=\frac{\pi}{3}\right)}=- \frac{9 \sqrt{3}}{2} + 3 \pi$$$

Answer: $$$\int_{\frac{\pi}{3}}^{\frac{\pi}{2}}\left( 36 \cos^{2}{\left(\theta \right)} \right)d\theta=- \frac{9 \sqrt{3}}{2} + 3 \pi\approx 1.63054932670943$$$

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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