Calculadora de integrais definidas e impróprias

Calcular integrais definidas e impróprias passo a passo

A calculadora tentará calcular a integral definida (ou seja, com limites), inclusive imprópria, com as etapas mostradas.

Enter a function:

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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Your input: calculate $$$\int_{0}^{2}\left( 3 x^{2} + x - 1 \right)dx$$$

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

$$$\left(x^{3} + \frac{x^{2}}{2} - x\right)|_{\left(x=2\right)}=8$$$

$$$\left(x^{3} + \frac{x^{2}}{2} - x\right)|_{\left(x=0\right)}=0$$$

$$$\int_{0}^{2}\left( 3 x^{2} + x - 1 \right)dx=\left(x^{3} + \frac{x^{2}}{2} - x\right)|_{\left(x=2\right)}-\left(x^{3} + \frac{x^{2}}{2} - x\right)|_{\left(x=0\right)}=8$$$

Answer: $$$\int_{0}^{2}\left( 3 x^{2} + x - 1 \right)dx=8$$$