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

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

$$$\left(\frac{2^{x}}{\ln{\left(2 \right)}}\right)|_{\left(x=1\right)}=\frac{2}{\ln{\left(2 \right)}}$$$

$$$\left(\frac{2^{x}}{\ln{\left(2 \right)}}\right)|_{\left(x=0\right)}=\frac{1}{\ln{\left(2 \right)}}$$$

$$$\int_{0}^{1}\left( 2^{x} \right)dx=\left(\frac{2^{x}}{\ln{\left(2 \right)}}\right)|_{\left(x=1\right)}-\left(\frac{2^{x}}{\ln{\left(2 \right)}}\right)|_{\left(x=0\right)}=\frac{1}{\ln{\left(2 \right)}}$$$

Answer: $$$\int_{0}^{1}\left( 2^{x} \right)dx=\frac{1}{\ln{\left(2 \right)}}\approx 1.44269504088896$$$

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Solution