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 $$$\int_{4}^{8}\left( 8 x^{2} + 5 x - 9 \right)dx$$$
First, calculate the corresponding indefinite integral: $$$\int{\left(8 x^{2} + 5 x - 9\right)d x}=\frac{x \left(16 x^{2} + 15 x - 54\right)}{6}$$$ (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.
$$$\left(\frac{x \left(16 x^{2} + 15 x - 54\right)}{6}\right)|_{\left(x=8\right)}=\frac{4360}{3}$$$
$$$\left(\frac{x \left(16 x^{2} + 15 x - 54\right)}{6}\right)|_{\left(x=4\right)}=\frac{524}{3}$$$
$$$\int_{4}^{8}\left( 8 x^{2} + 5 x - 9 \right)dx=\left(\frac{x \left(16 x^{2} + 15 x - 54\right)}{6}\right)|_{\left(x=8\right)}-\left(\frac{x \left(16 x^{2} + 15 x - 54\right)}{6}\right)|_{\left(x=4\right)}=\frac{3836}{3}$$$
Answer: $$$\int_{4}^{8}\left( 8 x^{2} + 5 x - 9 \right)dx=\frac{3836}{3}\approx 1278.66666666667$$$