定積分與廣義積分計算器
逐步計算定積分與瑕積分
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。
Solution
Your input: calculate $$$\int_{\frac{\pi}{3}}^{\frac{5 \pi}{3}}\left( 8 \cos^{2}{\left(x \right)} \right)dx$$$
First, calculate the corresponding indefinite integral: $$$\int{8 \cos^{2}{\left(x \right)} d x}=4 x + 2 \sin{\left(2 x \right)}$$$ (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.
$$$\left(4 x + 2 \sin{\left(2 x \right)}\right)|_{\left(x=\frac{5 \pi}{3}\right)}=- \sqrt{3} + \frac{20 \pi}{3}$$$
$$$\left(4 x + 2 \sin{\left(2 x \right)}\right)|_{\left(x=\frac{\pi}{3}\right)}=\sqrt{3} + \frac{4 \pi}{3}$$$
$$$\int_{\frac{\pi}{3}}^{\frac{5 \pi}{3}}\left( 8 \cos^{2}{\left(x \right)} \right)dx=\left(4 x + 2 \sin{\left(2 x \right)}\right)|_{\left(x=\frac{5 \pi}{3}\right)}-\left(4 x + 2 \sin{\left(2 x \right)}\right)|_{\left(x=\frac{\pi}{3}\right)}=- 2 \sqrt{3} + \frac{16 \pi}{3}$$$
Answer: $$$\int_{\frac{\pi}{3}}^{\frac{5 \pi}{3}}\left( 8 \cos^{2}{\left(x \right)} \right)dx=- 2 \sqrt{3} + \frac{16 \pi}{3}\approx 13.2910592040078$$$