Calcolatore di integrali definiti e impropri

Il calcolatore cercherà di valutare l'integrale definito (cioè con estremi), inclusi quelli impropri, mostrando i passaggi.

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Solution

Your input: calculate $$$\int_{0}^{1}\left( x \operatorname{atan}{\left(x \right)} \right)dx$$$

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

$$$\left(\frac{x^{2} \operatorname{atan}{\left(x \right)} - x + \operatorname{atan}{\left(x \right)}}{2}\right)|_{\left(x=1\right)}=- \frac{1}{2} + \frac{\pi}{4}$$$

$$$\left(\frac{x^{2} \operatorname{atan}{\left(x \right)} - x + \operatorname{atan}{\left(x \right)}}{2}\right)|_{\left(x=0\right)}=0$$$

$$$\int_{0}^{1}\left( x \operatorname{atan}{\left(x \right)} \right)dx=\left(\frac{x^{2} \operatorname{atan}{\left(x \right)} - x + \operatorname{atan}{\left(x \right)}}{2}\right)|_{\left(x=1\right)}-\left(\frac{x^{2} \operatorname{atan}{\left(x \right)} - x + \operatorname{atan}{\left(x \right)}}{2}\right)|_{\left(x=0\right)}=- \frac{1}{2} + \frac{\pi}{4}$$$

Answer: $$$\int_{0}^{1}\left( x \operatorname{atan}{\left(x \right)} \right)dx=- \frac{1}{2} + \frac{\pi}{4}\approx 0.285398163397448$$$

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Solution