Määrättyjen ja epäoleellisten integraalien laskin

Laskin yrittää laskea määrätyn (eli rajoilla varustetun) integraalin, mukaan lukien epäoleelliset tapaukset, ja näyttää välivaiheet.

Enter a function:

Integrate with respect to:

Enter a lower limit:

If you need `-oo`, type -inf.

Enter an upper limit:

If you need `oo`, type inf.

Please write without any differentials such as `dx`, `dy` etc.

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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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Määrättyjen ja epäoleellisten integraalien laskin

Laske määrättyjä ja epäoleellisia integraaleja askel askeleelta

Solution