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
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。
Solution
Your input: calculate $$$\int_{0}^{\frac{2}{3}}\left( \cos^{2}{\left(x \right)} - \cos^{2}{\left(2 x \right)} \right)dx$$$
First, calculate the corresponding indefinite integral: $$$\int{\left(\cos^{2}{\left(x \right)} - \cos^{2}{\left(2 x \right)}\right)d x}=\sin^{3}{\left(x \right)} \cos{\left(x \right)}$$$ (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.
$$$\left(\sin^{3}{\left(x \right)} \cos{\left(x \right)}\right)|_{\left(x=\frac{2}{3}\right)}=\sin^{3}{\left(\frac{2}{3} \right)} \cos{\left(\frac{2}{3} \right)}$$$
$$$\left(\sin^{3}{\left(x \right)} \cos{\left(x \right)}\right)|_{\left(x=0\right)}=0$$$
$$$\int_{0}^{\frac{2}{3}}\left( \cos^{2}{\left(x \right)} - \cos^{2}{\left(2 x \right)} \right)dx=\left(\sin^{3}{\left(x \right)} \cos{\left(x \right)}\right)|_{\left(x=\frac{2}{3}\right)}-\left(\sin^{3}{\left(x \right)} \cos{\left(x \right)}\right)|_{\left(x=0\right)}=\sin^{3}{\left(\frac{2}{3} \right)} \cos{\left(\frac{2}{3} \right)}$$$
Answer: $$$\int_{0}^{\frac{2}{3}}\left( \cos^{2}{\left(x \right)} - \cos^{2}{\left(2 x \right)} \right)dx=\sin^{3}{\left(\frac{2}{3} \right)} \cos{\left(\frac{2}{3} \right)}\approx 0.185825397011352$$$