Calculatrice d’intégrales définies et impropres

La calculatrice essaiera d'évaluer l'intégrale définie (c.-à-d. avec bornes), y compris les intégrales impropres, en affichant les étapes.

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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: $$$\int_{0}^{\frac{\pi}{2}}\left( \frac{\pi \cos{\left(x \right)}}{2 \sqrt{\sin{\left(x \right)}}} \right)dx=\pi\approx 3.14159265358979$$$

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Calculatrice d’intégrales définies et impropres

Calculez des intégrales définies et impropres étape par étape

Solution