Calculadora de integrais definidas e impróprias

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.

If the calculator did not compute something or you have identified an error, or you have a suggestion/feedback, please write it in the comments below.

Solution

Your input: calculate $$$\int_{- t}^{0}\left( z \left(1 + \frac{z}{t}\right) \right)dz$$$

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

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

$$$\left(\frac{z^{2} \left(\frac{t}{2} + \frac{z}{3}\right)}{t}\right)|_{\left(z=- t\right)}=\frac{t^{2}}{6}$$$

$$$\int_{- t}^{0}\left( z \left(1 + \frac{z}{t}\right) \right)dz=\left(\frac{z^{2} \left(\frac{t}{2} + \frac{z}{3}\right)}{t}\right)|_{\left(z=0\right)}-\left(\frac{z^{2} \left(\frac{t}{2} + \frac{z}{3}\right)}{t}\right)|_{\left(z=- t\right)}=- \frac{t^{2}}{6}$$$

Answer: $$$\int_{- t}^{0}\left( z \left(1 + \frac{z}{t}\right) \right)dz=- \frac{t^{2}}{6}$$$

Calculadora de integrais definidas e impróprias

Calcular integrais definidas e impróprias passo a passo

Solution