Integral dari $$$\frac{\sqrt{7} \sqrt{7 - x}}{7 \sqrt{x}}$$$

Temukan $$$\int \frac{\sqrt{7} \sqrt{7 - x}}{7 \sqrt{x}}\, dx$$$.

Masukan ditulis ulang: $$$\int{\frac{\sqrt{7} \sqrt{7 - x}}{7 \sqrt{x}} d x}=\int{\frac{\sqrt{49 - 7 x}}{7 \sqrt{x}} d x}$$$.

Sederhanakan integran:

$${\color{red}{\int{\frac{\sqrt{49 - 7 x}}{7 \sqrt{x}} d x}}} = {\color{red}{\int{\frac{\sqrt{7} \sqrt{7 - x}}{7 \sqrt{x}} d x}}}$$

Terapkan aturan pengali konstanta $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$ dengan $$$c=\frac{\sqrt{7}}{7}$$$ dan $$$f{\left(x \right)} = \frac{\sqrt{7 - x}}{\sqrt{x}}$$$:

$${\color{red}{\int{\frac{\sqrt{7} \sqrt{7 - x}}{7 \sqrt{x}} d x}}} = {\color{red}{\left(\frac{\sqrt{7} \int{\frac{\sqrt{7 - x}}{\sqrt{x}} d x}}{7}\right)}}$$

Misalkan $$$u=\sqrt{x}$$$.

Kemudian $$$du=\left(\sqrt{x}\right)^{\prime }dx = \frac{1}{2 \sqrt{x}} dx$$$ (langkah-langkah dapat dilihat di »), dan kita memperoleh $$$\frac{dx}{\sqrt{x}} = 2 du$$$.

Integralnya menjadi

$$\frac{\sqrt{7} {\color{red}{\int{\frac{\sqrt{7 - x}}{\sqrt{x}} d x}}}}{7} = \frac{\sqrt{7} {\color{red}{\int{2 \sqrt{7 - u^{2}} d u}}}}{7}$$

Terapkan aturan pengali konstanta $$$\int c f{\left(u \right)}\, du = c \int f{\left(u \right)}\, du$$$ dengan $$$c=2$$$ dan $$$f{\left(u \right)} = \sqrt{7 - u^{2}}$$$:

$$\frac{\sqrt{7} {\color{red}{\int{2 \sqrt{7 - u^{2}} d u}}}}{7} = \frac{\sqrt{7} {\color{red}{\left(2 \int{\sqrt{7 - u^{2}} d u}\right)}}}{7}$$

Misalkan $$$u=\sqrt{7} \sin{\left(v \right)}$$$.

Maka $$$du=\left(\sqrt{7} \sin{\left(v \right)}\right)^{\prime }dv = \sqrt{7} \cos{\left(v \right)} dv$$$ (langkah-langkah dapat dilihat »).

Selain itu, berlaku $$$v=\operatorname{asin}{\left(\frac{\sqrt{7} u}{7} \right)}$$$.

Jadi,

$$$\sqrt{7 - u ^{2}} = \sqrt{7 - 7 \sin^{2}{\left( v \right)}}$$$

Gunakan identitas $$$1 - \sin^{2}{\left( v \right)} = \cos^{2}{\left( v \right)}$$$:

$$$\sqrt{7 - 7 \sin^{2}{\left( v \right)}}=\sqrt{7} \sqrt{1 - \sin^{2}{\left( v \right)}}=\sqrt{7} \sqrt{\cos^{2}{\left( v \right)}}$$$

Dengan asumsi bahwa $$$\cos{\left( v \right)} \ge 0$$$, diperoleh sebagai berikut:

$$$\sqrt{7} \sqrt{\cos^{2}{\left( v \right)}} = \sqrt{7} \cos{\left( v \right)}$$$

Jadi,

$$\frac{2 \sqrt{7} {\color{red}{\int{\sqrt{7 - u^{2}} d u}}}}{7} = \frac{2 \sqrt{7} {\color{red}{\int{7 \cos^{2}{\left(v \right)} d v}}}}{7}$$

Terapkan aturan pengali konstanta $$$\int c f{\left(v \right)}\, dv = c \int f{\left(v \right)}\, dv$$$ dengan $$$c=7$$$ dan $$$f{\left(v \right)} = \cos^{2}{\left(v \right)}$$$:

$$\frac{2 \sqrt{7} {\color{red}{\int{7 \cos^{2}{\left(v \right)} d v}}}}{7} = \frac{2 \sqrt{7} {\color{red}{\left(7 \int{\cos^{2}{\left(v \right)} d v}\right)}}}{7}$$

Terapkan rumus reduksi pangkat $$$\cos^{2}{\left(\alpha \right)} = \frac{\cos{\left(2 \alpha \right)}}{2} + \frac{1}{2}$$$ dengan $$$\alpha= v $$$:

$$2 \sqrt{7} {\color{red}{\int{\cos^{2}{\left(v \right)} d v}}} = 2 \sqrt{7} {\color{red}{\int{\left(\frac{\cos{\left(2 v \right)}}{2} + \frac{1}{2}\right)d v}}}$$

Terapkan aturan pengali konstanta $$$\int c f{\left(v \right)}\, dv = c \int f{\left(v \right)}\, dv$$$ dengan $$$c=\frac{1}{2}$$$ dan $$$f{\left(v \right)} = \cos{\left(2 v \right)} + 1$$$:

$$2 \sqrt{7} {\color{red}{\int{\left(\frac{\cos{\left(2 v \right)}}{2} + \frac{1}{2}\right)d v}}} = 2 \sqrt{7} {\color{red}{\left(\frac{\int{\left(\cos{\left(2 v \right)} + 1\right)d v}}{2}\right)}}$$

Integralkan suku demi suku:

$$\sqrt{7} {\color{red}{\int{\left(\cos{\left(2 v \right)} + 1\right)d v}}} = \sqrt{7} {\color{red}{\left(\int{1 d v} + \int{\cos{\left(2 v \right)} d v}\right)}}$$

Terapkan aturan konstanta $$$\int c\, dv = c v$$$ dengan $$$c=1$$$:

$$\sqrt{7} \left(\int{\cos{\left(2 v \right)} d v} + {\color{red}{\int{1 d v}}}\right) = \sqrt{7} \left(\int{\cos{\left(2 v \right)} d v} + {\color{red}{v}}\right)$$

Misalkan $$$w=2 v$$$.

Kemudian $$$dw=\left(2 v\right)^{\prime }dv = 2 dv$$$ (langkah-langkah dapat dilihat di »), dan kita memperoleh $$$dv = \frac{dw}{2}$$$.

Integral tersebut dapat ditulis ulang sebagai

$$\sqrt{7} \left(v + {\color{red}{\int{\cos{\left(2 v \right)} d v}}}\right) = \sqrt{7} \left(v + {\color{red}{\int{\frac{\cos{\left(w \right)}}{2} d w}}}\right)$$

Terapkan aturan pengali konstanta $$$\int c f{\left(w \right)}\, dw = c \int f{\left(w \right)}\, dw$$$ dengan $$$c=\frac{1}{2}$$$ dan $$$f{\left(w \right)} = \cos{\left(w \right)}$$$:

$$\sqrt{7} \left(v + {\color{red}{\int{\frac{\cos{\left(w \right)}}{2} d w}}}\right) = \sqrt{7} \left(v + {\color{red}{\left(\frac{\int{\cos{\left(w \right)} d w}}{2}\right)}}\right)$$

Integral dari kosinus adalah $$$\int{\cos{\left(w \right)} d w} = \sin{\left(w \right)}$$$:

$$\sqrt{7} \left(v + \frac{{\color{red}{\int{\cos{\left(w \right)} d w}}}}{2}\right) = \sqrt{7} \left(v + \frac{{\color{red}{\sin{\left(w \right)}}}}{2}\right)$$

Ingat bahwa $$$w=2 v$$$:

$$\sqrt{7} \left(v + \frac{\sin{\left({\color{red}{w}} \right)}}{2}\right) = \sqrt{7} \left(v + \frac{\sin{\left({\color{red}{\left(2 v\right)}} \right)}}{2}\right)$$

Ingat bahwa $$$v=\operatorname{asin}{\left(\frac{\sqrt{7} u}{7} \right)}$$$:

$$\sqrt{7} \left(\frac{\sin{\left(2 {\color{red}{v}} \right)}}{2} + {\color{red}{v}}\right) = \sqrt{7} \left(\frac{\sin{\left(2 {\color{red}{\operatorname{asin}{\left(\frac{\sqrt{7} u}{7} \right)}}} \right)}}{2} + {\color{red}{\operatorname{asin}{\left(\frac{\sqrt{7} u}{7} \right)}}}\right)$$

Ingat bahwa $$$u=\sqrt{x}$$$:

$$\sqrt{7} \left(\frac{\sin{\left(2 \operatorname{asin}{\left(\frac{\sqrt{7} {\color{red}{u}}}{7} \right)} \right)}}{2} + \operatorname{asin}{\left(\frac{\sqrt{7} {\color{red}{u}}}{7} \right)}\right) = \sqrt{7} \left(\frac{\sin{\left(2 \operatorname{asin}{\left(\frac{\sqrt{7} {\color{red}{\sqrt{x}}}}{7} \right)} \right)}}{2} + \operatorname{asin}{\left(\frac{\sqrt{7} {\color{red}{\sqrt{x}}}}{7} \right)}\right)$$

Oleh karena itu,

$$\int{\frac{\sqrt{49 - 7 x}}{7 \sqrt{x}} d x} = \sqrt{7} \left(\frac{\sin{\left(2 \operatorname{asin}{\left(\frac{\sqrt{7} \sqrt{x}}{7} \right)} \right)}}{2} + \operatorname{asin}{\left(\frac{\sqrt{7} \sqrt{x}}{7} \right)}\right)$$

Dengan menggunakan rumus $$$\sin{\left(2 \operatorname{asin}{\left(\alpha \right)} \right)} = 2 \alpha \sqrt{1 - \alpha^{2}}$$$, $$$\sin{\left(2 \operatorname{acos}{\left(\alpha \right)} \right)} = 2 \alpha \sqrt{1 - \alpha^{2}}$$$, $$$\cos{\left(2 \operatorname{asin}{\left(\alpha \right)} \right)} = 1 - 2 \alpha^{2}$$$, $$$\cos{\left(2 \operatorname{acos}{\left(\alpha \right)} \right)} = 2 \alpha^{2} - 1$$$, $$$\sinh{\left(2 \operatorname{asinh}{\left(\alpha \right)} \right)} = 2 \alpha \sqrt{\alpha^{2} + 1}$$$, $$$\sinh{\left(2 \operatorname{acosh}{\left(\alpha \right)} \right)} = 2 \alpha \sqrt{\alpha - 1} \sqrt{\alpha + 1}$$$, $$$\cosh{\left(2 \operatorname{asinh}{\left(\alpha \right)} \right)} = 2 \alpha^{2} + 1$$$, $$$\cosh{\left(2 \operatorname{acosh}{\left(\alpha \right)} \right)} = 2 \alpha^{2} - 1$$$, sederhanakan ekspresi:

$$\int{\frac{\sqrt{49 - 7 x}}{7 \sqrt{x}} d x} = \sqrt{7} \left(\frac{\sqrt{7} \sqrt{x} \sqrt{1 - \frac{x}{7}}}{7} + \operatorname{asin}{\left(\frac{\sqrt{7} \sqrt{x}}{7} \right)}\right)$$

Sederhanakan lebih lanjut:

$$\int{\frac{\sqrt{49 - 7 x}}{7 \sqrt{x}} d x} = \sqrt{7} \left(\frac{\sqrt{x} \sqrt{7 - x}}{7} + \operatorname{asin}{\left(\frac{\sqrt{7} \sqrt{x}}{7} \right)}\right)$$

Tambahkan konstanta integrasi:

$$\int{\frac{\sqrt{49 - 7 x}}{7 \sqrt{x}} d x} = \sqrt{7} \left(\frac{\sqrt{x} \sqrt{7 - x}}{7} + \operatorname{asin}{\left(\frac{\sqrt{7} \sqrt{x}}{7} \right)}\right)+C$$

$$$\int \frac{\sqrt{7} \sqrt{7 - x}}{7 \sqrt{x}}\, dx = \sqrt{7} \left(\frac{\sqrt{x} \sqrt{7 - x}}{7} + \operatorname{asin}{\left(\frac{\sqrt{7} \sqrt{x}}{7} \right)}\right) + C$$$A