Integral dari $$$4 \tan{\left(3 x \right)}$$$

Temukan $$$\int 4 \tan{\left(3 x \right)}\, dx$$$.

Terapkan aturan pengali konstanta $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$ dengan $$$c=4$$$ dan $$$f{\left(x \right)} = \tan{\left(3 x \right)}$$$:

$${\color{red}{\int{4 \tan{\left(3 x \right)} d x}}} = {\color{red}{\left(4 \int{\tan{\left(3 x \right)} d x}\right)}}$$

Misalkan $$$u=3 x$$$.

Kemudian $$$du=\left(3 x\right)^{\prime }dx = 3 dx$$$ (langkah-langkah dapat dilihat di »), dan kita memperoleh $$$dx = \frac{du}{3}$$$.

Integralnya menjadi

$$4 {\color{red}{\int{\tan{\left(3 x \right)} d x}}} = 4 {\color{red}{\int{\frac{\tan{\left(u \right)}}{3} d u}}}$$

Terapkan aturan pengali konstanta $$$\int c f{\left(u \right)}\, du = c \int f{\left(u \right)}\, du$$$ dengan $$$c=\frac{1}{3}$$$ dan $$$f{\left(u \right)} = \tan{\left(u \right)}$$$:

$$4 {\color{red}{\int{\frac{\tan{\left(u \right)}}{3} d u}}} = 4 {\color{red}{\left(\frac{\int{\tan{\left(u \right)} d u}}{3}\right)}}$$

Tulis ulang tangen sebagai $$$\tan\left( u \right)=\frac{\sin\left( u \right)}{\cos\left( u \right)}$$$:

$$\frac{4 {\color{red}{\int{\tan{\left(u \right)} d u}}}}{3} = \frac{4 {\color{red}{\int{\frac{\sin{\left(u \right)}}{\cos{\left(u \right)}} d u}}}}{3}$$

Misalkan $$$v=\cos{\left(u \right)}$$$.

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

Dengan demikian,

$$\frac{4 {\color{red}{\int{\frac{\sin{\left(u \right)}}{\cos{\left(u \right)}} d u}}}}{3} = \frac{4 {\color{red}{\int{\left(- \frac{1}{v}\right)d v}}}}{3}$$

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

$$\frac{4 {\color{red}{\int{\left(- \frac{1}{v}\right)d v}}}}{3} = \frac{4 {\color{red}{\left(- \int{\frac{1}{v} d v}\right)}}}{3}$$

Integral dari $$$\frac{1}{v}$$$ adalah $$$\int{\frac{1}{v} d v} = \ln{\left(\left|{v}\right| \right)}$$$:

$$- \frac{4 {\color{red}{\int{\frac{1}{v} d v}}}}{3} = - \frac{4 {\color{red}{\ln{\left(\left|{v}\right| \right)}}}}{3}$$

Ingat bahwa $$$v=\cos{\left(u \right)}$$$:

$$- \frac{4 \ln{\left(\left|{{\color{red}{v}}}\right| \right)}}{3} = - \frac{4 \ln{\left(\left|{{\color{red}{\cos{\left(u \right)}}}}\right| \right)}}{3}$$

Ingat bahwa $$$u=3 x$$$:

$$- \frac{4 \ln{\left(\left|{\cos{\left({\color{red}{u}} \right)}}\right| \right)}}{3} = - \frac{4 \ln{\left(\left|{\cos{\left({\color{red}{\left(3 x\right)}} \right)}}\right| \right)}}{3}$$

Oleh karena itu,

$$\int{4 \tan{\left(3 x \right)} d x} = - \frac{4 \ln{\left(\left|{\cos{\left(3 x \right)}}\right| \right)}}{3}$$

Tambahkan konstanta integrasi:

$$\int{4 \tan{\left(3 x \right)} d x} = - \frac{4 \ln{\left(\left|{\cos{\left(3 x \right)}}\right| \right)}}{3}+C$$

$$$\int 4 \tan{\left(3 x \right)}\, dx = - \frac{4 \ln\left(\left|{\cos{\left(3 x \right)}}\right|\right)}{3} + C$$$A