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 $$$\int_{0}^{\frac{\pi}{4}}\left( \cos^{3}{\left(2 x \right)} \right)dx$$$
First, calculate the corresponding indefinite integral: $$$\int{\cos^{3}{\left(2 x \right)} d x}=\frac{\left(3 - \sin^{2}{\left(2 x \right)}\right) \sin{\left(2 x \right)}}{6}$$$ (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.
$$$\left(\frac{\left(3 - \sin^{2}{\left(2 x \right)}\right) \sin{\left(2 x \right)}}{6}\right)|_{\left(x=\frac{\pi}{4}\right)}=\frac{1}{3}$$$
$$$\left(\frac{\left(3 - \sin^{2}{\left(2 x \right)}\right) \sin{\left(2 x \right)}}{6}\right)|_{\left(x=0\right)}=0$$$
$$$\int_{0}^{\frac{\pi}{4}}\left( \cos^{3}{\left(2 x \right)} \right)dx=\left(\frac{\left(3 - \sin^{2}{\left(2 x \right)}\right) \sin{\left(2 x \right)}}{6}\right)|_{\left(x=\frac{\pi}{4}\right)}-\left(\frac{\left(3 - \sin^{2}{\left(2 x \right)}\right) \sin{\left(2 x \right)}}{6}\right)|_{\left(x=0\right)}=\frac{1}{3}$$$
Answer: $$$\int_{0}^{\frac{\pi}{4}}\left( \cos^{3}{\left(2 x \right)} \right)dx=\frac{1}{3}\approx 0.333333333333333$$$