Integral dari $$$e^{4 \theta} \sin{\left(5 \theta \right)}$$$

Temukan $$$\int e^{4 \theta} \sin{\left(5 \theta \right)}\, d\theta$$$.

Untuk integral $$$\int{e^{4 \theta} \sin{\left(5 \theta \right)} d \theta}$$$, gunakan integrasi parsial $$$\int \operatorname{u} \operatorname{dv} = \operatorname{u}\operatorname{v} - \int \operatorname{v} \operatorname{du}$$$.

Misalkan $$$\operatorname{u}=\sin{\left(5 \theta \right)}$$$ dan $$$\operatorname{dv}=e^{4 \theta} d\theta$$$.

Maka $$$\operatorname{du}=\left(\sin{\left(5 \theta \right)}\right)^{\prime }d\theta=5 \cos{\left(5 \theta \right)} d\theta$$$ (langkah-langkah dapat dilihat di ») dan $$$\operatorname{v}=\int{e^{4 \theta} d \theta}=\frac{e^{4 \theta}}{4}$$$ (langkah-langkah dapat dilihat di »).

Dengan demikian,

$${\color{red}{\int{e^{4 \theta} \sin{\left(5 \theta \right)} d \theta}}}={\color{red}{\left(\sin{\left(5 \theta \right)} \cdot \frac{e^{4 \theta}}{4}-\int{\frac{e^{4 \theta}}{4} \cdot 5 \cos{\left(5 \theta \right)} d \theta}\right)}}={\color{red}{\left(\frac{e^{4 \theta} \sin{\left(5 \theta \right)}}{4} - \int{\frac{5 e^{4 \theta} \cos{\left(5 \theta \right)}}{4} d \theta}\right)}}$$

Terapkan aturan pengali konstanta $$$\int c f{\left(\theta \right)}\, d\theta = c \int f{\left(\theta \right)}\, d\theta$$$ dengan $$$c=\frac{5}{4}$$$ dan $$$f{\left(\theta \right)} = e^{4 \theta} \cos{\left(5 \theta \right)}$$$:

$$\frac{e^{4 \theta} \sin{\left(5 \theta \right)}}{4} - {\color{red}{\int{\frac{5 e^{4 \theta} \cos{\left(5 \theta \right)}}{4} d \theta}}} = \frac{e^{4 \theta} \sin{\left(5 \theta \right)}}{4} - {\color{red}{\left(\frac{5 \int{e^{4 \theta} \cos{\left(5 \theta \right)} d \theta}}{4}\right)}}$$

Untuk integral $$$\int{e^{4 \theta} \cos{\left(5 \theta \right)} d \theta}$$$, gunakan integrasi parsial $$$\int \operatorname{u} \operatorname{dv} = \operatorname{u}\operatorname{v} - \int \operatorname{v} \operatorname{du}$$$.

Misalkan $$$\operatorname{u}=\cos{\left(5 \theta \right)}$$$ dan $$$\operatorname{dv}=e^{4 \theta} d\theta$$$.

Maka $$$\operatorname{du}=\left(\cos{\left(5 \theta \right)}\right)^{\prime }d\theta=- 5 \sin{\left(5 \theta \right)} d\theta$$$ (langkah-langkah dapat dilihat di ») dan $$$\operatorname{v}=\int{e^{4 \theta} d \theta}=\frac{e^{4 \theta}}{4}$$$ (langkah-langkah dapat dilihat di »).

Integralnya menjadi

$$\frac{e^{4 \theta} \sin{\left(5 \theta \right)}}{4} - \frac{5 {\color{red}{\int{e^{4 \theta} \cos{\left(5 \theta \right)} d \theta}}}}{4}=\frac{e^{4 \theta} \sin{\left(5 \theta \right)}}{4} - \frac{5 {\color{red}{\left(\cos{\left(5 \theta \right)} \cdot \frac{e^{4 \theta}}{4}-\int{\frac{e^{4 \theta}}{4} \cdot \left(- 5 \sin{\left(5 \theta \right)}\right) d \theta}\right)}}}{4}=\frac{e^{4 \theta} \sin{\left(5 \theta \right)}}{4} - \frac{5 {\color{red}{\left(\frac{e^{4 \theta} \cos{\left(5 \theta \right)}}{4} - \int{\left(- \frac{5 e^{4 \theta} \sin{\left(5 \theta \right)}}{4}\right)d \theta}\right)}}}{4}$$

Terapkan aturan pengali konstanta $$$\int c f{\left(\theta \right)}\, d\theta = c \int f{\left(\theta \right)}\, d\theta$$$ dengan $$$c=- \frac{5}{4}$$$ dan $$$f{\left(\theta \right)} = e^{4 \theta} \sin{\left(5 \theta \right)}$$$:

$$\frac{e^{4 \theta} \sin{\left(5 \theta \right)}}{4} - \frac{5 e^{4 \theta} \cos{\left(5 \theta \right)}}{16} + \frac{5 {\color{red}{\int{\left(- \frac{5 e^{4 \theta} \sin{\left(5 \theta \right)}}{4}\right)d \theta}}}}{4} = \frac{e^{4 \theta} \sin{\left(5 \theta \right)}}{4} - \frac{5 e^{4 \theta} \cos{\left(5 \theta \right)}}{16} + \frac{5 {\color{red}{\left(- \frac{5 \int{e^{4 \theta} \sin{\left(5 \theta \right)} d \theta}}{4}\right)}}}{4}$$

Kita telah sampai pada integral yang sudah pernah kita lihat.

Dengan demikian, kita telah memperoleh persamaan sederhana berikut sehubungan dengan integral:

$$\int{e^{4 \theta} \sin{\left(5 \theta \right)} d \theta} = \frac{e^{4 \theta} \sin{\left(5 \theta \right)}}{4} - \frac{5 e^{4 \theta} \cos{\left(5 \theta \right)}}{16} - \frac{25 \int{e^{4 \theta} \sin{\left(5 \theta \right)} d \theta}}{16}$$

Dengan menyelesaikannya, kita memperoleh bahwa

$$\int{e^{4 \theta} \sin{\left(5 \theta \right)} d \theta} = \frac{\left(4 \sin{\left(5 \theta \right)} - 5 \cos{\left(5 \theta \right)}\right) e^{4 \theta}}{41}$$

Oleh karena itu,

$$\int{e^{4 \theta} \sin{\left(5 \theta \right)} d \theta} = \frac{\left(4 \sin{\left(5 \theta \right)} - 5 \cos{\left(5 \theta \right)}\right) e^{4 \theta}}{41}$$

Tambahkan konstanta integrasi:

$$\int{e^{4 \theta} \sin{\left(5 \theta \right)} d \theta} = \frac{\left(4 \sin{\left(5 \theta \right)} - 5 \cos{\left(5 \theta \right)}\right) e^{4 \theta}}{41}+C$$

$$$\int e^{4 \theta} \sin{\left(5 \theta \right)}\, d\theta = \frac{\left(4 \sin{\left(5 \theta \right)} - 5 \cos{\left(5 \theta \right)}\right) e^{4 \theta}}{41} + C$$$A