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Definite and Improper Integral Calculator
Calculate definite and improper integrals step by step
The calculator will try to evaluate the definite (i.e. with bounds) integral, including improper, with steps shown.
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: $$$\int_{0}^{\frac{\pi}{2}}\left( \frac{\pi \cos{\left(x \right)}}{2 \sqrt{\sin{\left(x \right)}}} \right)dx=\pi\approx 3.14159265358979$$$