Calcolatore di integrali definiti e impropri
Calcola integrali definiti e impropri passo dopo passo
Il calcolatore cercherà di valutare l'integrale definito (cioè con estremi), inclusi quelli impropri, mostrando i passaggi.
Solution
Your input: calculate $$$\int_{0}^{1}\left( \ln{\left(x^{2} + 1 \right)} \right)dx$$$
First, calculate the corresponding indefinite integral: $$$\int{\ln{\left(x^{2} + 1 \right)} d x}=x \ln{\left(x^{2} + 1 \right)} - 2 x + 2 \operatorname{atan}{\left(x \right)}$$$ (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.
$$$\left(x \ln{\left(x^{2} + 1 \right)} - 2 x + 2 \operatorname{atan}{\left(x \right)}\right)|_{\left(x=1\right)}=-2 + \ln{\left(2 \right)} + \frac{\pi}{2}$$$
$$$\left(x \ln{\left(x^{2} + 1 \right)} - 2 x + 2 \operatorname{atan}{\left(x \right)}\right)|_{\left(x=0\right)}=0$$$
$$$\int_{0}^{1}\left( \ln{\left(x^{2} + 1 \right)} \right)dx=\left(x \ln{\left(x^{2} + 1 \right)} - 2 x + 2 \operatorname{atan}{\left(x \right)}\right)|_{\left(x=1\right)}-\left(x \ln{\left(x^{2} + 1 \right)} - 2 x + 2 \operatorname{atan}{\left(x \right)}\right)|_{\left(x=0\right)}=-2 + \ln{\left(2 \right)} + \frac{\pi}{2}$$$
Answer: $$$\int_{0}^{1}\left( \ln{\left(x^{2} + 1 \right)} \right)dx=-2 + \ln{\left(2 \right)} + \frac{\pi}{2}\approx 0.263943507354842$$$