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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 $$$\int_{0}^{1}\left( x^{2} \left(2 x^{3} + 3\right)^{3} \right)dx$$$
First, calculate the corresponding indefinite integral: $$$\int{x^{2} \left(2 x^{3} + 3\right)^{3} d x}=\frac{\left(2 x^{3} + 3\right)^{4}}{24}$$$ (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.
$$$\left(\frac{\left(2 x^{3} + 3\right)^{4}}{24}\right)|_{\left(x=1\right)}=\frac{625}{24}$$$
$$$\left(\frac{\left(2 x^{3} + 3\right)^{4}}{24}\right)|_{\left(x=0\right)}=\frac{27}{8}$$$
$$$\int_{0}^{1}\left( x^{2} \left(2 x^{3} + 3\right)^{3} \right)dx=\left(\frac{\left(2 x^{3} + 3\right)^{4}}{24}\right)|_{\left(x=1\right)}-\left(\frac{\left(2 x^{3} + 3\right)^{4}}{24}\right)|_{\left(x=0\right)}=\frac{68}{3}$$$
Answer: $$$\int_{0}^{1}\left( x^{2} \left(2 x^{3} + 3\right)^{3} \right)dx=\frac{68}{3}\approx 22.6666666666667$$$