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Calculadora de integrales definidas e impropias
Calcular integrales definidas e impropias paso a paso
La calculadora intentará evaluar la integral definida (es decir, con límites de integración), incluyendo las impropias, mostrando los pasos.
Solution
Your input: calculate $$$\int_{1}^{2}\left( 4 x^{3} - x \right)dx$$$
First, calculate the corresponding indefinite integral: $$$\int{\left(4 x^{3} - x\right)d x}=x^{4} - \frac{x^{2}}{2}$$$ (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.
$$$\left(x^{4} - \frac{x^{2}}{2}\right)|_{\left(x=2\right)}=14$$$
$$$\left(x^{4} - \frac{x^{2}}{2}\right)|_{\left(x=1\right)}=\frac{1}{2}$$$
$$$\int_{1}^{2}\left( 4 x^{3} - x \right)dx=\left(x^{4} - \frac{x^{2}}{2}\right)|_{\left(x=2\right)}-\left(x^{4} - \frac{x^{2}}{2}\right)|_{\left(x=1\right)}=\frac{27}{2}$$$
Answer: $$$\int_{1}^{2}\left( 4 x^{3} - x \right)dx=\frac{27}{2}=13.5$$$