Calculadora de integrales definidas e impropias
Calcular integrales definidas e impropias paso a paso
La calculadora intentará evaluar la integral definida (es decir, con límites de integración), incluyendo las impropias, mostrando los pasos.
Solution
Your input: calculate $$$\int_{2}^{3}\left( \sqrt{x - 2} + 1 \right)dx$$$
First, calculate the corresponding indefinite integral: $$$\int{\left(\sqrt{x - 2} + 1\right)d x}=x + \frac{2 \left(x - 2\right)^{\frac{3}{2}}}{3}$$$ (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.
$$$\left(x + \frac{2 \left(x - 2\right)^{\frac{3}{2}}}{3}\right)|_{\left(x=3\right)}=\frac{11}{3}$$$
$$$\left(x + \frac{2 \left(x - 2\right)^{\frac{3}{2}}}{3}\right)|_{\left(x=2\right)}=2$$$
$$$\int_{2}^{3}\left( \sqrt{x - 2} + 1 \right)dx=\left(x + \frac{2 \left(x - 2\right)^{\frac{3}{2}}}{3}\right)|_{\left(x=3\right)}-\left(x + \frac{2 \left(x - 2\right)^{\frac{3}{2}}}{3}\right)|_{\left(x=2\right)}=\frac{5}{3}$$$
Answer: $$$\int_{2}^{3}\left( \sqrt{x - 2} + 1 \right)dx=\frac{5}{3}\approx 1.66666666666667$$$