$$$\pi$$$ değişkenine göre $$$\frac{\sin^{2}{\left(z \right)}}{- \frac{\pi}{6} + z}$$$ fonksiyonunun integrali

Bulun: $$$\int \frac{\sin^{2}{\left(z \right)}}{- \frac{\pi}{6} + z}\, d\pi$$$.

Sabit katsayı kuralı $$$\int c f{\left(\pi \right)}\, d\pi = c \int f{\left(\pi \right)}\, d\pi$$$'i $$$c=\sin^{2}{\left(z \right)}$$$ ve $$$f{\left(\pi \right)} = \frac{1}{- \frac{\pi}{6} + z}$$$ ile uygula:

$${\color{red}{\int{\frac{\sin^{2}{\left(z \right)}}{- \frac{\pi}{6} + z} d \pi}}} = {\color{red}{\sin^{2}{\left(z \right)} \int{\frac{1}{- \frac{\pi}{6} + z} d \pi}}}$$

$$$u=- \frac{\pi}{6} + z$$$ olsun.

Böylece $$$du=\left(- \frac{\pi}{6} + z\right)^{\prime }d\pi = - \frac{d\pi}{6}$$$ (adımlar » görülebilir) ve $$$d\pi = - 6 du$$$ elde ederiz.

Dolayısıyla,

$$\sin^{2}{\left(z \right)} {\color{red}{\int{\frac{1}{- \frac{\pi}{6} + z} d \pi}}} = \sin^{2}{\left(z \right)} {\color{red}{\int{\left(- \frac{6}{u}\right)d u}}}$$

Sabit katsayı kuralı $$$\int c f{\left(u \right)}\, du = c \int f{\left(u \right)}\, du$$$'i $$$c=-6$$$ ve $$$f{\left(u \right)} = \frac{1}{u}$$$ ile uygula:

$$\sin^{2}{\left(z \right)} {\color{red}{\int{\left(- \frac{6}{u}\right)d u}}} = \sin^{2}{\left(z \right)} {\color{red}{\left(- 6 \int{\frac{1}{u} d u}\right)}}$$

$$$\frac{1}{u}$$$'nin integrali $$$\int{\frac{1}{u} d u} = \ln{\left(\left|{u}\right| \right)}$$$:

$$- 6 \sin^{2}{\left(z \right)} {\color{red}{\int{\frac{1}{u} d u}}} = - 6 \sin^{2}{\left(z \right)} {\color{red}{\ln{\left(\left|{u}\right| \right)}}}$$

Hatırlayın ki $$$u=- \frac{\pi}{6} + z$$$:

$$- 6 \ln{\left(\left|{{\color{red}{u}}}\right| \right)} \sin^{2}{\left(z \right)} = - 6 \ln{\left(\left|{{\color{red}{\left(- \frac{\pi}{6} + z\right)}}}\right| \right)} \sin^{2}{\left(z \right)}$$

Dolayısıyla,

$$\int{\frac{\sin^{2}{\left(z \right)}}{- \frac{\pi}{6} + z} d \pi} = - 6 \ln{\left(\left|{\frac{\pi}{6} - z}\right| \right)} \sin^{2}{\left(z \right)}$$

Sadeleştirin:

$$\int{\frac{\sin^{2}{\left(z \right)}}{- \frac{\pi}{6} + z} d \pi} = 6 \left(- \ln{\left(\left|{\pi - 6 z}\right| \right)} + \ln{\left(6 \right)}\right) \sin^{2}{\left(z \right)}$$

İntegrasyon sabitini ekleyin:

$$\int{\frac{\sin^{2}{\left(z \right)}}{- \frac{\pi}{6} + z} d \pi} = 6 \left(- \ln{\left(\left|{\pi - 6 z}\right| \right)} + \ln{\left(6 \right)}\right) \sin^{2}{\left(z \right)}+C$$

$$$\int \frac{\sin^{2}{\left(z \right)}}{- \frac{\pi}{6} + z}\, d\pi = 6 \left(- \ln\left(\left|{\pi - 6 z}\right|\right) + \ln\left(6\right)\right) \sin^{2}{\left(z \right)} + C$$$A