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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 $$$\int_{0}^{\frac{\pi}{2}}\left( \frac{\pi \cos{\left(x \right)}}{2 \sqrt{\sin{\left(x \right)}}} \right)dx$$$
First, calculate the corresponding indefinite integral: $$$\int{\frac{\pi \cos{\left(x \right)}}{2 \sqrt{\sin{\left(x \right)}}} d x}=\pi \sqrt{\sin{\left(x \right)}}$$$ (for steps, see indefinite integral calculator)
The interval of integration contains the point $$$0$$$, which is not in the domain of the integrand, so this is an improper integral of type 2.
To evaluate an integral over an interval, we use the Fundamental Theorem of Calculus. However, we need to use limit if an endpoint of the interval is special (is not in the domain of the function).
$$$\int_{0}^{\frac{\pi}{2}}\left( \frac{\pi \cos{\left(x \right)}}{2 \sqrt{\sin{\left(x \right)}}} \right)dx=\left(\pi \sqrt{\sin{\left(x \right)}}\right)|_{\left(x=\frac{\pi}{2}\right)}-\lim_{x \to 0}\left(\pi \sqrt{\sin{\left(x \right)}}\right)=\pi$$$
Answer: $$$\int_{0}^{\frac{\pi}{2}}\left( \frac{\pi \cos{\left(x \right)}}{2 \sqrt{\sin{\left(x \right)}}} \right)dx=\pi\approx 3.14159265358979$$$