Integral de $$$\frac{x + 1}{\sqrt{2 x - 1}}$$$

Encontre $$$\int \frac{x + 1}{\sqrt{2 x - 1}}\, dx$$$.

Reescreva o numerador como $$$x + 1=\frac{2 x - 1}{2} + \frac{3}{2}$$$ e separe a fração:

$${\color{red}{\int{\frac{x + 1}{\sqrt{2 x - 1}} d x}}} = {\color{red}{\int{\left(\frac{\sqrt{2 x - 1}}{2} + \frac{3}{2 \sqrt{2 x - 1}}\right)d x}}}$$

Integre termo a termo:

$${\color{red}{\int{\left(\frac{\sqrt{2 x - 1}}{2} + \frac{3}{2 \sqrt{2 x - 1}}\right)d x}}} = {\color{red}{\left(\int{\frac{3}{2 \sqrt{2 x - 1}} d x} + \int{\frac{\sqrt{2 x - 1}}{2} d x}\right)}}$$

Aplique a regra do múltiplo constante $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$ usando $$$c=\frac{1}{2}$$$ e $$$f{\left(x \right)} = \sqrt{2 x - 1}$$$:

$$\int{\frac{3}{2 \sqrt{2 x - 1}} d x} + {\color{red}{\int{\frac{\sqrt{2 x - 1}}{2} d x}}} = \int{\frac{3}{2 \sqrt{2 x - 1}} d x} + {\color{red}{\left(\frac{\int{\sqrt{2 x - 1} d x}}{2}\right)}}$$

Seja $$$u=2 x - 1$$$.

Então $$$du=\left(2 x - 1\right)^{\prime }dx = 2 dx$$$ (veja os passos »), e obtemos $$$dx = \frac{du}{2}$$$.

Portanto,

$$\int{\frac{3}{2 \sqrt{2 x - 1}} d x} + \frac{{\color{red}{\int{\sqrt{2 x - 1} d x}}}}{2} = \int{\frac{3}{2 \sqrt{2 x - 1}} d x} + \frac{{\color{red}{\int{\frac{\sqrt{u}}{2} d u}}}}{2}$$

Aplique a regra do múltiplo constante $$$\int c f{\left(u \right)}\, du = c \int f{\left(u \right)}\, du$$$ usando $$$c=\frac{1}{2}$$$ e $$$f{\left(u \right)} = \sqrt{u}$$$:

$$\int{\frac{3}{2 \sqrt{2 x - 1}} d x} + \frac{{\color{red}{\int{\frac{\sqrt{u}}{2} d u}}}}{2} = \int{\frac{3}{2 \sqrt{2 x - 1}} d x} + \frac{{\color{red}{\left(\frac{\int{\sqrt{u} d u}}{2}\right)}}}{2}$$

Aplique a regra da potência $$$\int u^{n}\, du = \frac{u^{n + 1}}{n + 1}$$$ $$$\left(n \neq -1 \right)$$$ com $$$n=\frac{1}{2}$$$:

$$\int{\frac{3}{2 \sqrt{2 x - 1}} d x} + \frac{{\color{red}{\int{\sqrt{u} d u}}}}{4}=\int{\frac{3}{2 \sqrt{2 x - 1}} d x} + \frac{{\color{red}{\int{u^{\frac{1}{2}} d u}}}}{4}=\int{\frac{3}{2 \sqrt{2 x - 1}} d x} + \frac{{\color{red}{\frac{u^{\frac{1}{2} + 1}}{\frac{1}{2} + 1}}}}{4}=\int{\frac{3}{2 \sqrt{2 x - 1}} d x} + \frac{{\color{red}{\left(\frac{2 u^{\frac{3}{2}}}{3}\right)}}}{4}$$

Recorde que $$$u=2 x - 1$$$:

$$\int{\frac{3}{2 \sqrt{2 x - 1}} d x} + \frac{{\color{red}{u}}^{\frac{3}{2}}}{6} = \int{\frac{3}{2 \sqrt{2 x - 1}} d x} + \frac{{\color{red}{\left(2 x - 1\right)}}^{\frac{3}{2}}}{6}$$

Aplique a regra do múltiplo constante $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$ usando $$$c=\frac{3}{2}$$$ e $$$f{\left(x \right)} = \frac{1}{\sqrt{2 x - 1}}$$$:

$$\frac{\left(2 x - 1\right)^{\frac{3}{2}}}{6} + {\color{red}{\int{\frac{3}{2 \sqrt{2 x - 1}} d x}}} = \frac{\left(2 x - 1\right)^{\frac{3}{2}}}{6} + {\color{red}{\left(\frac{3 \int{\frac{1}{\sqrt{2 x - 1}} d x}}{2}\right)}}$$

Seja $$$u=2 x - 1$$$.

Então $$$du=\left(2 x - 1\right)^{\prime }dx = 2 dx$$$ (veja os passos »), e obtemos $$$dx = \frac{du}{2}$$$.

A integral pode ser reescrita como

$$\frac{\left(2 x - 1\right)^{\frac{3}{2}}}{6} + \frac{3 {\color{red}{\int{\frac{1}{\sqrt{2 x - 1}} d x}}}}{2} = \frac{\left(2 x - 1\right)^{\frac{3}{2}}}{6} + \frac{3 {\color{red}{\int{\frac{1}{2 \sqrt{u}} d u}}}}{2}$$

Aplique a regra do múltiplo constante $$$\int c f{\left(u \right)}\, du = c \int f{\left(u \right)}\, du$$$ usando $$$c=\frac{1}{2}$$$ e $$$f{\left(u \right)} = \frac{1}{\sqrt{u}}$$$:

$$\frac{\left(2 x - 1\right)^{\frac{3}{2}}}{6} + \frac{3 {\color{red}{\int{\frac{1}{2 \sqrt{u}} d u}}}}{2} = \frac{\left(2 x - 1\right)^{\frac{3}{2}}}{6} + \frac{3 {\color{red}{\left(\frac{\int{\frac{1}{\sqrt{u}} d u}}{2}\right)}}}{2}$$

Aplique a regra da potência $$$\int u^{n}\, du = \frac{u^{n + 1}}{n + 1}$$$ $$$\left(n \neq -1 \right)$$$ com $$$n=- \frac{1}{2}$$$:

$$\frac{\left(2 x - 1\right)^{\frac{3}{2}}}{6} + \frac{3 {\color{red}{\int{\frac{1}{\sqrt{u}} d u}}}}{4}=\frac{\left(2 x - 1\right)^{\frac{3}{2}}}{6} + \frac{3 {\color{red}{\int{u^{- \frac{1}{2}} d u}}}}{4}=\frac{\left(2 x - 1\right)^{\frac{3}{2}}}{6} + \frac{3 {\color{red}{\frac{u^{- \frac{1}{2} + 1}}{- \frac{1}{2} + 1}}}}{4}=\frac{\left(2 x - 1\right)^{\frac{3}{2}}}{6} + \frac{3 {\color{red}{\left(2 u^{\frac{1}{2}}\right)}}}{4}=\frac{\left(2 x - 1\right)^{\frac{3}{2}}}{6} + \frac{3 {\color{red}{\left(2 \sqrt{u}\right)}}}{4}$$

Recorde que $$$u=2 x - 1$$$:

$$\frac{\left(2 x - 1\right)^{\frac{3}{2}}}{6} + \frac{3 \sqrt{{\color{red}{u}}}}{2} = \frac{\left(2 x - 1\right)^{\frac{3}{2}}}{6} + \frac{3 \sqrt{{\color{red}{\left(2 x - 1\right)}}}}{2}$$

Portanto,

$$\int{\frac{x + 1}{\sqrt{2 x - 1}} d x} = \frac{\left(2 x - 1\right)^{\frac{3}{2}}}{6} + \frac{3 \sqrt{2 x - 1}}{2}$$

Simplifique:

$$\int{\frac{x + 1}{\sqrt{2 x - 1}} d x} = \frac{\left(x + 4\right) \sqrt{2 x - 1}}{3}$$

Adicione a constante de integração:

$$\int{\frac{x + 1}{\sqrt{2 x - 1}} d x} = \frac{\left(x + 4\right) \sqrt{2 x - 1}}{3}+C$$

$$$\int \frac{x + 1}{\sqrt{2 x - 1}}\, dx = \frac{\left(x + 4\right) \sqrt{2 x - 1}}{3} + C$$$A