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 $$$\int_{0}^{e}\left( e^{\frac{x}{2}} - 2 \right)dx$$$
First, calculate the corresponding indefinite integral: $$$\int{\left(e^{\frac{x}{2}} - 2\right)d x}=- 2 x + 2 e^{\frac{x}{2}}$$$ (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.
$$$\left(- 2 x + 2 e^{\frac{x}{2}}\right)|_{\left(x=e\right)}=- 2 e + 2 e^{\frac{e}{2}}$$$
$$$\left(- 2 x + 2 e^{\frac{x}{2}}\right)|_{\left(x=0\right)}=2$$$
$$$\int_{0}^{e}\left( e^{\frac{x}{2}} - 2 \right)dx=\left(- 2 x + 2 e^{\frac{x}{2}}\right)|_{\left(x=e\right)}-\left(- 2 x + 2 e^{\frac{x}{2}}\right)|_{\left(x=0\right)}=- 2 e - 2 + 2 e^{\frac{e}{2}}$$$
Answer: $$$\int_{0}^{e}\left( e^{\frac{x}{2}} - 2 \right)dx=- 2 e - 2 + 2 e^{\frac{e}{2}}\approx 0.349131492901035$$$