Calculadora de Integrais Definidas e Impróprias

A calculadora tentará calcular a integral definida (isto é, com limites), inclusive no caso impróprio, com os passos mostrados.

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Integrate with respect to:

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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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Solution

Your input: calculate $$$\int_{0}^{9}\left( 9 e^{\sqrt{x}} \right)dx$$$

First, calculate the corresponding indefinite integral: $$$\int{9 e^{\sqrt{x}} d x}=18 \left(\sqrt{x} - 1\right) e^{\sqrt{x}}$$$ (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.

$$$\left(18 \left(\sqrt{x} - 1\right) e^{\sqrt{x}}\right)|_{\left(x=9\right)}=36 e^{3}$$$

$$$\left(18 \left(\sqrt{x} - 1\right) e^{\sqrt{x}}\right)|_{\left(x=0\right)}=-18$$$

$$$\int_{0}^{9}\left( 9 e^{\sqrt{x}} \right)dx=\left(18 \left(\sqrt{x} - 1\right) e^{\sqrt{x}}\right)|_{\left(x=9\right)}-\left(18 \left(\sqrt{x} - 1\right) e^{\sqrt{x}}\right)|_{\left(x=0\right)}=18 + 36 e^{3}$$$

Answer: $$$\int_{0}^{9}\left( 9 e^{\sqrt{x}} \right)dx=18 + 36 e^{3}\approx 741.079329234756$$$

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Calculadora de Integrais Definidas e Impróprias

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Solution