Määrättyjen ja epäoleellisten integraalien laskin

Laskin yrittää laskea määrätyn (eli rajoilla varustetun) integraalin, mukaan lukien epäoleelliset tapaukset, ja näyttää välivaiheet.

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Solution

Your input: calculate $$$\int_{0}^{-1}\left( - 4 x^{3} + x^{2} \right)dx$$$

First, calculate the corresponding indefinite integral: $$$\int{\left(- 4 x^{3} + x^{2}\right)d x}=x^{3} \left(\frac{1}{3} - x\right)$$$ (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.

$$$\left(x^{3} \left(\frac{1}{3} - x\right)\right)|_{\left(x=-1\right)}=- \frac{4}{3}$$$

$$$\left(x^{3} \left(\frac{1}{3} - x\right)\right)|_{\left(x=0\right)}=0$$$

$$$\int_{0}^{-1}\left( - 4 x^{3} + x^{2} \right)dx=\left(x^{3} \left(\frac{1}{3} - x\right)\right)|_{\left(x=-1\right)}-\left(x^{3} \left(\frac{1}{3} - x\right)\right)|_{\left(x=0\right)}=- \frac{4}{3}$$$

Answer: $$$\int_{0}^{-1}\left( - 4 x^{3} + x^{2} \right)dx=- \frac{4}{3}\approx -1.33333333333333$$$

Määrättyjen ja epäoleellisten integraalien laskin

Laske määrättyjä ja epäoleellisia integraaleja askel askeleelta

Solution