定积分与广义积分计算器

该计算器将尝试计算定积分（即带上下限的积分），包括广义积分，并显示求解步骤。

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Solution

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

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

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

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

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

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

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定积分与广义积分计算器

逐步计算定积分和广义积分

Solution