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Integral de $$$x^{2} - \frac{1}{\sqrt{2 x - 1}}$$$
Calculadora relacionada: Calculadora de integrales definidas e impropias
Tu entrada
Halla $$$\int \left(x^{2} - \frac{1}{\sqrt{2 x - 1}}\right)\, dx$$$.
Solución
Integra término a término:
$${\color{red}{\int{\left(x^{2} - \frac{1}{\sqrt{2 x - 1}}\right)d x}}} = {\color{red}{\left(\int{x^{2} d x} - \int{\frac{1}{\sqrt{2 x - 1}} d x}\right)}}$$
Aplica la regla de la potencia $$$\int x^{n}\, dx = \frac{x^{n + 1}}{n + 1}$$$ $$$\left(n \neq -1 \right)$$$ con $$$n=2$$$:
$$- \int{\frac{1}{\sqrt{2 x - 1}} d x} + {\color{red}{\int{x^{2} d x}}}=- \int{\frac{1}{\sqrt{2 x - 1}} d x} + {\color{red}{\frac{x^{1 + 2}}{1 + 2}}}=- \int{\frac{1}{\sqrt{2 x - 1}} d x} + {\color{red}{\left(\frac{x^{3}}{3}\right)}}$$
Sea $$$u=2 x - 1$$$.
Entonces $$$du=\left(2 x - 1\right)^{\prime }dx = 2 dx$$$ (los pasos pueden verse »), y obtenemos que $$$dx = \frac{du}{2}$$$.
Entonces,
$$\frac{x^{3}}{3} - {\color{red}{\int{\frac{1}{\sqrt{2 x - 1}} d x}}} = \frac{x^{3}}{3} - {\color{red}{\int{\frac{1}{2 \sqrt{u}} d u}}}$$
Aplica la regla del factor constante $$$\int c f{\left(u \right)}\, du = c \int f{\left(u \right)}\, du$$$ con $$$c=\frac{1}{2}$$$ y $$$f{\left(u \right)} = \frac{1}{\sqrt{u}}$$$:
$$\frac{x^{3}}{3} - {\color{red}{\int{\frac{1}{2 \sqrt{u}} d u}}} = \frac{x^{3}}{3} - {\color{red}{\left(\frac{\int{\frac{1}{\sqrt{u}} d u}}{2}\right)}}$$
Aplica la regla de la potencia $$$\int u^{n}\, du = \frac{u^{n + 1}}{n + 1}$$$ $$$\left(n \neq -1 \right)$$$ con $$$n=- \frac{1}{2}$$$:
$$\frac{x^{3}}{3} - \frac{{\color{red}{\int{\frac{1}{\sqrt{u}} d u}}}}{2}=\frac{x^{3}}{3} - \frac{{\color{red}{\int{u^{- \frac{1}{2}} d u}}}}{2}=\frac{x^{3}}{3} - \frac{{\color{red}{\frac{u^{- \frac{1}{2} + 1}}{- \frac{1}{2} + 1}}}}{2}=\frac{x^{3}}{3} - \frac{{\color{red}{\left(2 u^{\frac{1}{2}}\right)}}}{2}=\frac{x^{3}}{3} - \frac{{\color{red}{\left(2 \sqrt{u}\right)}}}{2}$$
Recordemos que $$$u=2 x - 1$$$:
$$\frac{x^{3}}{3} - \sqrt{{\color{red}{u}}} = \frac{x^{3}}{3} - \sqrt{{\color{red}{\left(2 x - 1\right)}}}$$
Por lo tanto,
$$\int{\left(x^{2} - \frac{1}{\sqrt{2 x - 1}}\right)d x} = \frac{x^{3}}{3} - \sqrt{2 x - 1}$$
Añade la constante de integración:
$$\int{\left(x^{2} - \frac{1}{\sqrt{2 x - 1}}\right)d x} = \frac{x^{3}}{3} - \sqrt{2 x - 1}+C$$
Respuesta
$$$\int \left(x^{2} - \frac{1}{\sqrt{2 x - 1}}\right)\, dx = \left(\frac{x^{3}}{3} - \sqrt{2 x - 1}\right) + C$$$A