Calculadora de integrales definidas e impropias

La calculadora intentará evaluar la integral definida (es decir, con límites de integración), incluyendo las impropias, mostrando los pasos.

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Solution

Your input: calculate $$$\int_{-1}^{1}\left( x^{2} + \operatorname{asin}{\left(x \right)} \right)dx$$$

First, calculate the corresponding indefinite integral: $$$\int{\left(x^{2} + \operatorname{asin}{\left(x \right)}\right)d x}=\frac{x^{3}}{3} + x \operatorname{asin}{\left(x \right)} + \sqrt{1 - x^{2}}$$$ (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.

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

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

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

Answer: $$$\int_{-1}^{1}\left( x^{2} + \operatorname{asin}{\left(x \right)} \right)dx=\frac{2}{3}\approx 0.666666666666667$$$

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Solution