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 $$$\int_{\frac{1}{3}}^{\frac{3}{4}}\left( 6 x \left(1 - x\right) \right)dx$$$
First, calculate the corresponding indefinite integral: $$$\int{6 x \left(1 - x\right) d x}=x^{2} \left(3 - 2 x\right)$$$ (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.
$$$\left(x^{2} \left(3 - 2 x\right)\right)|_{\left(x=\frac{3}{4}\right)}=\frac{27}{32}$$$
$$$\left(x^{2} \left(3 - 2 x\right)\right)|_{\left(x=\frac{1}{3}\right)}=\frac{7}{27}$$$
$$$\int_{\frac{1}{3}}^{\frac{3}{4}}\left( 6 x \left(1 - x\right) \right)dx=\left(x^{2} \left(3 - 2 x\right)\right)|_{\left(x=\frac{3}{4}\right)}-\left(x^{2} \left(3 - 2 x\right)\right)|_{\left(x=\frac{1}{3}\right)}=\frac{505}{864}$$$
Answer: $$$\int_{\frac{1}{3}}^{\frac{3}{4}}\left( 6 x \left(1 - x\right) \right)dx=\frac{505}{864}\approx 0.584490740740741$$$