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Calculadora de Integrais Definidas e Impróprias
Calcule integrais definidas e impróprias passo a passo
A calculadora tentará calcular a integral definida (isto é, com limites), inclusive no caso impróprio, com os passos mostrados.
Solution
Your input: calculate $$$\int_{- \frac{\pi}{2}}^{\frac{\pi}{2}}\left( e^{x} \cos{\left(x \right)} \right)dx$$$
First, calculate the corresponding indefinite integral: $$$\int{e^{x} \cos{\left(x \right)} d x}=\frac{\sqrt{2} e^{x} \sin{\left(x + \frac{\pi}{4} \right)}}{2}$$$ (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.
$$$\left(\frac{\sqrt{2} e^{x} \sin{\left(x + \frac{\pi}{4} \right)}}{2}\right)|_{\left(x=\frac{\pi}{2}\right)}=\frac{e^{\frac{\pi}{2}}}{2}$$$
$$$\left(\frac{\sqrt{2} e^{x} \sin{\left(x + \frac{\pi}{4} \right)}}{2}\right)|_{\left(x=- \frac{\pi}{2}\right)}=- \frac{1}{2 e^{\frac{\pi}{2}}}$$$
$$$\int_{- \frac{\pi}{2}}^{\frac{\pi}{2}}\left( e^{x} \cos{\left(x \right)} \right)dx=\left(\frac{\sqrt{2} e^{x} \sin{\left(x + \frac{\pi}{4} \right)}}{2}\right)|_{\left(x=\frac{\pi}{2}\right)}-\left(\frac{\sqrt{2} e^{x} \sin{\left(x + \frac{\pi}{4} \right)}}{2}\right)|_{\left(x=- \frac{\pi}{2}\right)}=\frac{1}{2 e^{\frac{\pi}{2}}} + \frac{e^{\frac{\pi}{2}}}{2}$$$
Answer: $$$\int_{- \frac{\pi}{2}}^{\frac{\pi}{2}}\left( e^{x} \cos{\left(x \right)} \right)dx=\frac{1}{2 e^{\frac{\pi}{2}}} + \frac{e^{\frac{\pi}{2}}}{2}\approx 2.50917847865806$$$