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Integral dari $$$\frac{x \sin{\left(x \right)}}{2}$$$
Kalkulator terkait: Kalkulator Integral Tentu dan Tak Wajar
Masukan Anda
Temukan $$$\int \frac{x \sin{\left(x \right)}}{2}\, dx$$$.
Solusi
Terapkan aturan pengali konstanta $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$ dengan $$$c=\frac{1}{2}$$$ dan $$$f{\left(x \right)} = x \sin{\left(x \right)}$$$:
$${\color{red}{\int{\frac{x \sin{\left(x \right)}}{2} d x}}} = {\color{red}{\left(\frac{\int{x \sin{\left(x \right)} d x}}{2}\right)}}$$
Untuk integral $$$\int{x \sin{\left(x \right)} d x}$$$, gunakan integrasi parsial $$$\int \operatorname{u} \operatorname{dv} = \operatorname{u}\operatorname{v} - \int \operatorname{v} \operatorname{du}$$$.
Misalkan $$$\operatorname{u}=x$$$ dan $$$\operatorname{dv}=\sin{\left(x \right)} dx$$$.
Maka $$$\operatorname{du}=\left(x\right)^{\prime }dx=1 dx$$$ (langkah-langkah dapat dilihat di ») dan $$$\operatorname{v}=\int{\sin{\left(x \right)} d x}=- \cos{\left(x \right)}$$$ (langkah-langkah dapat dilihat di »).
Jadi,
$$\frac{{\color{red}{\int{x \sin{\left(x \right)} d x}}}}{2}=\frac{{\color{red}{\left(x \cdot \left(- \cos{\left(x \right)}\right)-\int{\left(- \cos{\left(x \right)}\right) \cdot 1 d x}\right)}}}{2}=\frac{{\color{red}{\left(- x \cos{\left(x \right)} - \int{\left(- \cos{\left(x \right)}\right)d x}\right)}}}{2}$$
Terapkan aturan pengali konstanta $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$ dengan $$$c=-1$$$ dan $$$f{\left(x \right)} = \cos{\left(x \right)}$$$:
$$- \frac{x \cos{\left(x \right)}}{2} - \frac{{\color{red}{\int{\left(- \cos{\left(x \right)}\right)d x}}}}{2} = - \frac{x \cos{\left(x \right)}}{2} - \frac{{\color{red}{\left(- \int{\cos{\left(x \right)} d x}\right)}}}{2}$$
Integral dari kosinus adalah $$$\int{\cos{\left(x \right)} d x} = \sin{\left(x \right)}$$$:
$$- \frac{x \cos{\left(x \right)}}{2} + \frac{{\color{red}{\int{\cos{\left(x \right)} d x}}}}{2} = - \frac{x \cos{\left(x \right)}}{2} + \frac{{\color{red}{\sin{\left(x \right)}}}}{2}$$
Oleh karena itu,
$$\int{\frac{x \sin{\left(x \right)}}{2} d x} = - \frac{x \cos{\left(x \right)}}{2} + \frac{\sin{\left(x \right)}}{2}$$
Tambahkan konstanta integrasi:
$$\int{\frac{x \sin{\left(x \right)}}{2} d x} = - \frac{x \cos{\left(x \right)}}{2} + \frac{\sin{\left(x \right)}}{2}+C$$
Jawaban
$$$\int \frac{x \sin{\left(x \right)}}{2}\, dx = \left(- \frac{x \cos{\left(x \right)}}{2} + \frac{\sin{\left(x \right)}}{2}\right) + C$$$A