Calculadora de Integrais Definidas e Impróprias

A calculadora tentará calcular a integral definida (isto é, com limites), inclusive no caso impróprio, com os passos mostrados.

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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 $$$\int_{0}^{\frac{1}{2}}\left( \operatorname{asin}{\left(x \right)} \right)dx$$$

First, calculate the corresponding indefinite integral: $$$\int{\operatorname{asin}{\left(x \right)} d x}=x \operatorname{asin}{\left(x \right)} + \sqrt{1 - x^{2}}$$$ (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.

$$$\left(x \operatorname{asin}{\left(x \right)} + \sqrt{1 - x^{2}}\right)|_{\left(x=\frac{1}{2}\right)}=\frac{\pi}{12} + \frac{\sqrt{3}}{2}$$$

$$$\left(x \operatorname{asin}{\left(x \right)} + \sqrt{1 - x^{2}}\right)|_{\left(x=0\right)}=1$$$

$$$\int_{0}^{\frac{1}{2}}\left( \operatorname{asin}{\left(x \right)} \right)dx=\left(x \operatorname{asin}{\left(x \right)} + \sqrt{1 - x^{2}}\right)|_{\left(x=\frac{1}{2}\right)}-\left(x \operatorname{asin}{\left(x \right)} + \sqrt{1 - x^{2}}\right)|_{\left(x=0\right)}=-1 + \frac{\pi}{12} + \frac{\sqrt{3}}{2}$$$

Answer: $$$\int_{0}^{\frac{1}{2}}\left( \operatorname{asin}{\left(x \right)} \right)dx=-1 + \frac{\pi}{12} + \frac{\sqrt{3}}{2}\approx 0.127824791583588$$$

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Calculadora de Integrais Definidas e Impróprias

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Solution