Integral dari $$$- \operatorname{acos}{\left(5 x \right)}$$$

Temukan $$$\int \left(- \operatorname{acos}{\left(5 x \right)}\right)\, dx$$$.

Terapkan aturan pengali konstanta $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$ dengan $$$c=-1$$$ dan $$$f{\left(x \right)} = \operatorname{acos}{\left(5 x \right)}$$$:

$${\color{red}{\int{\left(- \operatorname{acos}{\left(5 x \right)}\right)d x}}} = {\color{red}{\left(- \int{\operatorname{acos}{\left(5 x \right)} d x}\right)}}$$

Misalkan $$$u=5 x$$$.

Kemudian $$$du=\left(5 x\right)^{\prime }dx = 5 dx$$$ (langkah-langkah dapat dilihat di »), dan kita memperoleh $$$dx = \frac{du}{5}$$$.

Oleh karena itu,

$$- {\color{red}{\int{\operatorname{acos}{\left(5 x \right)} d x}}} = - {\color{red}{\int{\frac{\operatorname{acos}{\left(u \right)}}{5} d u}}}$$

Terapkan aturan pengali konstanta $$$\int c f{\left(u \right)}\, du = c \int f{\left(u \right)}\, du$$$ dengan $$$c=\frac{1}{5}$$$ dan $$$f{\left(u \right)} = \operatorname{acos}{\left(u \right)}$$$:

$$- {\color{red}{\int{\frac{\operatorname{acos}{\left(u \right)}}{5} d u}}} = - {\color{red}{\left(\frac{\int{\operatorname{acos}{\left(u \right)} d u}}{5}\right)}}$$

Untuk integral $$$\int{\operatorname{acos}{\left(u \right)} d u}$$$, gunakan integrasi parsial $$$\int \operatorname{\kappa} \operatorname{dv} = \operatorname{\kappa}\operatorname{v} - \int \operatorname{v} \operatorname{d\kappa}$$$.

Misalkan $$$\operatorname{\kappa}=\operatorname{acos}{\left(u \right)}$$$ dan $$$\operatorname{dv}=du$$$.

Maka $$$\operatorname{d\kappa}=\left(\operatorname{acos}{\left(u \right)}\right)^{\prime }du=- \frac{1}{\sqrt{1 - u^{2}}} du$$$ (langkah-langkah dapat dilihat di ») dan $$$\operatorname{v}=\int{1 d u}=u$$$ (langkah-langkah dapat dilihat di »).

Jadi,

$$- \frac{{\color{red}{\int{\operatorname{acos}{\left(u \right)} d u}}}}{5}=- \frac{{\color{red}{\left(\operatorname{acos}{\left(u \right)} \cdot u-\int{u \cdot \left(- \frac{1}{\sqrt{1 - u^{2}}}\right) d u}\right)}}}{5}=- \frac{{\color{red}{\left(u \operatorname{acos}{\left(u \right)} - \int{\left(- \frac{u}{\sqrt{1 - u^{2}}}\right)d u}\right)}}}{5}$$

Misalkan $$$v=1 - u^{2}$$$.

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

Dengan demikian,

$$- \frac{u \operatorname{acos}{\left(u \right)}}{5} + \frac{{\color{red}{\int{\left(- \frac{u}{\sqrt{1 - u^{2}}}\right)d u}}}}{5} = - \frac{u \operatorname{acos}{\left(u \right)}}{5} + \frac{{\color{red}{\int{\frac{1}{2 \sqrt{v}} d v}}}}{5}$$

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

$$- \frac{u \operatorname{acos}{\left(u \right)}}{5} + \frac{{\color{red}{\int{\frac{1}{2 \sqrt{v}} d v}}}}{5} = - \frac{u \operatorname{acos}{\left(u \right)}}{5} + \frac{{\color{red}{\left(\frac{\int{\frac{1}{\sqrt{v}} d v}}{2}\right)}}}{5}$$

Terapkan aturan pangkat $$$\int v^{n}\, dv = \frac{v^{n + 1}}{n + 1}$$$ $$$\left(n \neq -1 \right)$$$ dengan $$$n=- \frac{1}{2}$$$:

$$- \frac{u \operatorname{acos}{\left(u \right)}}{5} + \frac{{\color{red}{\int{\frac{1}{\sqrt{v}} d v}}}}{10}=- \frac{u \operatorname{acos}{\left(u \right)}}{5} + \frac{{\color{red}{\int{v^{- \frac{1}{2}} d v}}}}{10}=- \frac{u \operatorname{acos}{\left(u \right)}}{5} + \frac{{\color{red}{\frac{v^{- \frac{1}{2} + 1}}{- \frac{1}{2} + 1}}}}{10}=- \frac{u \operatorname{acos}{\left(u \right)}}{5} + \frac{{\color{red}{\left(2 v^{\frac{1}{2}}\right)}}}{10}=- \frac{u \operatorname{acos}{\left(u \right)}}{5} + \frac{{\color{red}{\left(2 \sqrt{v}\right)}}}{10}$$

Ingat bahwa $$$v=1 - u^{2}$$$:

$$- \frac{u \operatorname{acos}{\left(u \right)}}{5} + \frac{\sqrt{{\color{red}{v}}}}{5} = - \frac{u \operatorname{acos}{\left(u \right)}}{5} + \frac{\sqrt{{\color{red}{\left(1 - u^{2}\right)}}}}{5}$$

Ingat bahwa $$$u=5 x$$$:

$$\frac{\sqrt{1 - {\color{red}{u}}^{2}}}{5} - \frac{{\color{red}{u}} \operatorname{acos}{\left({\color{red}{u}} \right)}}{5} = \frac{\sqrt{1 - {\color{red}{\left(5 x\right)}}^{2}}}{5} - \frac{{\color{red}{\left(5 x\right)}} \operatorname{acos}{\left({\color{red}{\left(5 x\right)}} \right)}}{5}$$

Oleh karena itu,

$$\int{\left(- \operatorname{acos}{\left(5 x \right)}\right)d x} = - x \operatorname{acos}{\left(5 x \right)} + \frac{\sqrt{1 - 25 x^{2}}}{5}$$

Tambahkan konstanta integrasi:

$$\int{\left(- \operatorname{acos}{\left(5 x \right)}\right)d x} = - x \operatorname{acos}{\left(5 x \right)} + \frac{\sqrt{1 - 25 x^{2}}}{5}+C$$

$$$\int \left(- \operatorname{acos}{\left(5 x \right)}\right)\, dx = \left(- x \operatorname{acos}{\left(5 x \right)} + \frac{\sqrt{1 - 25 x^{2}}}{5}\right) + C$$$A