$$$\frac{\pi \sin{\left(2 t \right)} \sin{\left(4 t \right)} \cos{\left(4 t \right)}}{20}$$$'nin integrali

Bulun: $$$\int \frac{\pi \sin{\left(2 t \right)} \sin{\left(4 t \right)} \cos{\left(4 t \right)}}{20}\, dt$$$.

$$$\sin\left(2 t \right)\cos\left(4 t \right)$$$ ifadesini $$$\alpha=2 t$$$ ve $$$\beta=4 t$$$ ile $$$\sin\left(\alpha \right)\cos\left(\beta \right)=\frac{1}{2} \sin\left(\alpha-\beta \right)+\frac{1}{2} \sin\left(\alpha+\beta \right)$$$ formülünü kullanarak yeniden yazın:

$${\color{red}{\int{\frac{\pi \sin{\left(2 t \right)} \sin{\left(4 t \right)} \cos{\left(4 t \right)}}{20} d t}}} = {\color{red}{\int{\frac{\pi \left(- \frac{\sin{\left(2 t \right)}}{2} + \frac{\sin{\left(6 t \right)}}{2}\right) \sin{\left(4 t \right)}}{20} d t}}}$$

İfadeyi genişletin:

$${\color{red}{\int{\frac{\pi \left(- \frac{\sin{\left(2 t \right)}}{2} + \frac{\sin{\left(6 t \right)}}{2}\right) \sin{\left(4 t \right)}}{20} d t}}} = {\color{red}{\int{\left(- \frac{\pi \sin{\left(2 t \right)} \sin{\left(4 t \right)}}{40} + \frac{\pi \sin{\left(4 t \right)} \sin{\left(6 t \right)}}{40}\right)d t}}}$$

Sabit katsayı kuralı $$$\int c f{\left(t \right)}\, dt = c \int f{\left(t \right)}\, dt$$$'i $$$c=\frac{1}{2}$$$ ve $$$f{\left(t \right)} = - \frac{\pi \sin{\left(2 t \right)} \sin{\left(4 t \right)}}{20} + \frac{\pi \sin{\left(4 t \right)} \sin{\left(6 t \right)}}{20}$$$ ile uygula:

$${\color{red}{\int{\left(- \frac{\pi \sin{\left(2 t \right)} \sin{\left(4 t \right)}}{40} + \frac{\pi \sin{\left(4 t \right)} \sin{\left(6 t \right)}}{40}\right)d t}}} = {\color{red}{\left(\frac{\int{\left(- \frac{\pi \sin{\left(2 t \right)} \sin{\left(4 t \right)}}{20} + \frac{\pi \sin{\left(4 t \right)} \sin{\left(6 t \right)}}{20}\right)d t}}{2}\right)}}$$

Her terimin integralini alın:

$$\frac{{\color{red}{\int{\left(- \frac{\pi \sin{\left(2 t \right)} \sin{\left(4 t \right)}}{20} + \frac{\pi \sin{\left(4 t \right)} \sin{\left(6 t \right)}}{20}\right)d t}}}}{2} = \frac{{\color{red}{\left(- \int{\frac{\pi \sin{\left(2 t \right)} \sin{\left(4 t \right)}}{20} d t} + \int{\frac{\pi \sin{\left(4 t \right)} \sin{\left(6 t \right)}}{20} d t}\right)}}}{2}$$

$$$\sin\left(2 t \right)\sin\left(4 t \right)$$$ ifadesini $$$\alpha=2 t$$$ ve $$$\beta=4 t$$$ ile $$$\sin\left(\alpha \right)\sin\left(\beta \right)=\frac{1}{2} \cos\left(\alpha-\beta \right)-\frac{1}{2} \cos\left(\alpha+\beta \right)$$$ formülünü kullanarak yeniden yazın:

$$\frac{\int{\frac{\pi \sin{\left(4 t \right)} \sin{\left(6 t \right)}}{20} d t}}{2} - \frac{{\color{red}{\int{\frac{\pi \sin{\left(2 t \right)} \sin{\left(4 t \right)}}{20} d t}}}}{2} = \frac{\int{\frac{\pi \sin{\left(4 t \right)} \sin{\left(6 t \right)}}{20} d t}}{2} - \frac{{\color{red}{\int{\frac{\pi \left(\frac{\cos{\left(2 t \right)}}{2} - \frac{\cos{\left(6 t \right)}}{2}\right)}{20} d t}}}}{2}$$

İfadeyi genişletin:

$$\frac{\int{\frac{\pi \sin{\left(4 t \right)} \sin{\left(6 t \right)}}{20} d t}}{2} - \frac{{\color{red}{\int{\frac{\pi \left(\frac{\cos{\left(2 t \right)}}{2} - \frac{\cos{\left(6 t \right)}}{2}\right)}{20} d t}}}}{2} = \frac{\int{\frac{\pi \sin{\left(4 t \right)} \sin{\left(6 t \right)}}{20} d t}}{2} - \frac{{\color{red}{\int{\left(\frac{\pi \cos{\left(2 t \right)}}{40} - \frac{\pi \cos{\left(6 t \right)}}{40}\right)d t}}}}{2}$$

Sabit katsayı kuralı $$$\int c f{\left(t \right)}\, dt = c \int f{\left(t \right)}\, dt$$$'i $$$c=\frac{1}{2}$$$ ve $$$f{\left(t \right)} = \frac{\pi \cos{\left(2 t \right)}}{20} - \frac{\pi \cos{\left(6 t \right)}}{20}$$$ ile uygula:

$$\frac{\int{\frac{\pi \sin{\left(4 t \right)} \sin{\left(6 t \right)}}{20} d t}}{2} - \frac{{\color{red}{\int{\left(\frac{\pi \cos{\left(2 t \right)}}{40} - \frac{\pi \cos{\left(6 t \right)}}{40}\right)d t}}}}{2} = \frac{\int{\frac{\pi \sin{\left(4 t \right)} \sin{\left(6 t \right)}}{20} d t}}{2} - \frac{{\color{red}{\left(\frac{\int{\left(\frac{\pi \cos{\left(2 t \right)}}{20} - \frac{\pi \cos{\left(6 t \right)}}{20}\right)d t}}{2}\right)}}}{2}$$

Her terimin integralini alın:

$$\frac{\int{\frac{\pi \sin{\left(4 t \right)} \sin{\left(6 t \right)}}{20} d t}}{2} - \frac{{\color{red}{\int{\left(\frac{\pi \cos{\left(2 t \right)}}{20} - \frac{\pi \cos{\left(6 t \right)}}{20}\right)d t}}}}{4} = \frac{\int{\frac{\pi \sin{\left(4 t \right)} \sin{\left(6 t \right)}}{20} d t}}{2} - \frac{{\color{red}{\left(\int{\frac{\pi \cos{\left(2 t \right)}}{20} d t} - \int{\frac{\pi \cos{\left(6 t \right)}}{20} d t}\right)}}}{4}$$

Sabit katsayı kuralı $$$\int c f{\left(t \right)}\, dt = c \int f{\left(t \right)}\, dt$$$'i $$$c=\frac{\pi}{20}$$$ ve $$$f{\left(t \right)} = \cos{\left(6 t \right)}$$$ ile uygula:

$$- \frac{\int{\frac{\pi \cos{\left(2 t \right)}}{20} d t}}{4} + \frac{\int{\frac{\pi \sin{\left(4 t \right)} \sin{\left(6 t \right)}}{20} d t}}{2} + \frac{{\color{red}{\int{\frac{\pi \cos{\left(6 t \right)}}{20} d t}}}}{4} = - \frac{\int{\frac{\pi \cos{\left(2 t \right)}}{20} d t}}{4} + \frac{\int{\frac{\pi \sin{\left(4 t \right)} \sin{\left(6 t \right)}}{20} d t}}{2} + \frac{{\color{red}{\left(\frac{\pi \int{\cos{\left(6 t \right)} d t}}{20}\right)}}}{4}$$

$$$u=6 t$$$ olsun.

Böylece $$$du=\left(6 t\right)^{\prime }dt = 6 dt$$$ (adımlar » görülebilir) ve $$$dt = \frac{du}{6}$$$ elde ederiz.

Dolayısıyla,

$$- \frac{\int{\frac{\pi \cos{\left(2 t \right)}}{20} d t}}{4} + \frac{\int{\frac{\pi \sin{\left(4 t \right)} \sin{\left(6 t \right)}}{20} d t}}{2} + \frac{\pi {\color{red}{\int{\cos{\left(6 t \right)} d t}}}}{80} = - \frac{\int{\frac{\pi \cos{\left(2 t \right)}}{20} d t}}{4} + \frac{\int{\frac{\pi \sin{\left(4 t \right)} \sin{\left(6 t \right)}}{20} d t}}{2} + \frac{\pi {\color{red}{\int{\frac{\cos{\left(u \right)}}{6} d u}}}}{80}$$

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

$$- \frac{\int{\frac{\pi \cos{\left(2 t \right)}}{20} d t}}{4} + \frac{\int{\frac{\pi \sin{\left(4 t \right)} \sin{\left(6 t \right)}}{20} d t}}{2} + \frac{\pi {\color{red}{\int{\frac{\cos{\left(u \right)}}{6} d u}}}}{80} = - \frac{\int{\frac{\pi \cos{\left(2 t \right)}}{20} d t}}{4} + \frac{\int{\frac{\pi \sin{\left(4 t \right)} \sin{\left(6 t \right)}}{20} d t}}{2} + \frac{\pi {\color{red}{\left(\frac{\int{\cos{\left(u \right)} d u}}{6}\right)}}}{80}$$

Kosinüsün integrali $$$\int{\cos{\left(u \right)} d u} = \sin{\left(u \right)}$$$:

$$- \frac{\int{\frac{\pi \cos{\left(2 t \right)}}{20} d t}}{4} + \frac{\int{\frac{\pi \sin{\left(4 t \right)} \sin{\left(6 t \right)}}{20} d t}}{2} + \frac{\pi {\color{red}{\int{\cos{\left(u \right)} d u}}}}{480} = - \frac{\int{\frac{\pi \cos{\left(2 t \right)}}{20} d t}}{4} + \frac{\int{\frac{\pi \sin{\left(4 t \right)} \sin{\left(6 t \right)}}{20} d t}}{2} + \frac{\pi {\color{red}{\sin{\left(u \right)}}}}{480}$$

Hatırlayın ki $$$u=6 t$$$:

$$- \frac{\int{\frac{\pi \cos{\left(2 t \right)}}{20} d t}}{4} + \frac{\int{\frac{\pi \sin{\left(4 t \right)} \sin{\left(6 t \right)}}{20} d t}}{2} + \frac{\pi \sin{\left({\color{red}{u}} \right)}}{480} = - \frac{\int{\frac{\pi \cos{\left(2 t \right)}}{20} d t}}{4} + \frac{\int{\frac{\pi \sin{\left(4 t \right)} \sin{\left(6 t \right)}}{20} d t}}{2} + \frac{\pi \sin{\left({\color{red}{\left(6 t\right)}} \right)}}{480}$$

Sabit katsayı kuralı $$$\int c f{\left(t \right)}\, dt = c \int f{\left(t \right)}\, dt$$$'i $$$c=\frac{\pi}{20}$$$ ve $$$f{\left(t \right)} = \cos{\left(2 t \right)}$$$ ile uygula:

$$\frac{\pi \sin{\left(6 t \right)}}{480} + \frac{\int{\frac{\pi \sin{\left(4 t \right)} \sin{\left(6 t \right)}}{20} d t}}{2} - \frac{{\color{red}{\int{\frac{\pi \cos{\left(2 t \right)}}{20} d t}}}}{4} = \frac{\pi \sin{\left(6 t \right)}}{480} + \frac{\int{\frac{\pi \sin{\left(4 t \right)} \sin{\left(6 t \right)}}{20} d t}}{2} - \frac{{\color{red}{\left(\frac{\pi \int{\cos{\left(2 t \right)} d t}}{20}\right)}}}{4}$$

$$$u=2 t$$$ olsun.

Böylece $$$du=\left(2 t\right)^{\prime }dt = 2 dt$$$ (adımlar » görülebilir) ve $$$dt = \frac{du}{2}$$$ elde ederiz.

Dolayısıyla,

$$\frac{\pi \sin{\left(6 t \right)}}{480} + \frac{\int{\frac{\pi \sin{\left(4 t \right)} \sin{\left(6 t \right)}}{20} d t}}{2} - \frac{\pi {\color{red}{\int{\cos{\left(2 t \right)} d t}}}}{80} = \frac{\pi \sin{\left(6 t \right)}}{480} + \frac{\int{\frac{\pi \sin{\left(4 t \right)} \sin{\left(6 t \right)}}{20} d t}}{2} - \frac{\pi {\color{red}{\int{\frac{\cos{\left(u \right)}}{2} d u}}}}{80}$$

Sabit katsayı kuralı $$$\int c f{\left(u \right)}\, du = c \int f{\left(u \right)}\, du$$$'i $$$c=\frac{1}{2}$$$ ve $$$f{\left(u \right)} = \cos{\left(u \right)}$$$ ile uygula:

$$\frac{\pi \sin{\left(6 t \right)}}{480} + \frac{\int{\frac{\pi \sin{\left(4 t \right)} \sin{\left(6 t \right)}}{20} d t}}{2} - \frac{\pi {\color{red}{\int{\frac{\cos{\left(u \right)}}{2} d u}}}}{80} = \frac{\pi \sin{\left(6 t \right)}}{480} + \frac{\int{\frac{\pi \sin{\left(4 t \right)} \sin{\left(6 t \right)}}{20} d t}}{2} - \frac{\pi {\color{red}{\left(\frac{\int{\cos{\left(u \right)} d u}}{2}\right)}}}{80}$$

Kosinüsün integrali $$$\int{\cos{\left(u \right)} d u} = \sin{\left(u \right)}$$$:

$$\frac{\pi \sin{\left(6 t \right)}}{480} + \frac{\int{\frac{\pi \sin{\left(4 t \right)} \sin{\left(6 t \right)}}{20} d t}}{2} - \frac{\pi {\color{red}{\int{\cos{\left(u \right)} d u}}}}{160} = \frac{\pi \sin{\left(6 t \right)}}{480} + \frac{\int{\frac{\pi \sin{\left(4 t \right)} \sin{\left(6 t \right)}}{20} d t}}{2} - \frac{\pi {\color{red}{\sin{\left(u \right)}}}}{160}$$

Hatırlayın ki $$$u=2 t$$$:

$$\frac{\pi \sin{\left(6 t \right)}}{480} + \frac{\int{\frac{\pi \sin{\left(4 t \right)} \sin{\left(6 t \right)}}{20} d t}}{2} - \frac{\pi \sin{\left({\color{red}{u}} \right)}}{160} = \frac{\pi \sin{\left(6 t \right)}}{480} + \frac{\int{\frac{\pi \sin{\left(4 t \right)} \sin{\left(6 t \right)}}{20} d t}}{2} - \frac{\pi \sin{\left({\color{red}{\left(2 t\right)}} \right)}}{160}$$

$$$\sin\left(4 t \right)\sin\left(6 t \right)$$$ ifadesini $$$\alpha=4 t$$$ ve $$$\beta=6 t$$$ ile $$$\sin\left(\alpha \right)\sin\left(\beta \right)=\frac{1}{2} \cos\left(\alpha-\beta \right)-\frac{1}{2} \cos\left(\alpha+\beta \right)$$$ formülünü kullanarak yeniden yazın:

$$- \frac{\pi \sin{\left(2 t \right)}}{160} + \frac{\pi \sin{\left(6 t \right)}}{480} + \frac{{\color{red}{\int{\frac{\pi \sin{\left(4 t \right)} \sin{\left(6 t \right)}}{20} d t}}}}{2} = - \frac{\pi \sin{\left(2 t \right)}}{160} + \frac{\pi \sin{\left(6 t \right)}}{480} + \frac{{\color{red}{\int{\frac{\pi \left(\frac{\cos{\left(2 t \right)}}{2} - \frac{\cos{\left(10 t \right)}}{2}\right)}{20} d t}}}}{2}$$

İfadeyi genişletin:

$$- \frac{\pi \sin{\left(2 t \right)}}{160} + \frac{\pi \sin{\left(6 t \right)}}{480} + \frac{{\color{red}{\int{\frac{\pi \left(\frac{\cos{\left(2 t \right)}}{2} - \frac{\cos{\left(10 t \right)}}{2}\right)}{20} d t}}}}{2} = - \frac{\pi \sin{\left(2 t \right)}}{160} + \frac{\pi \sin{\left(6 t \right)}}{480} + \frac{{\color{red}{\int{\left(\frac{\pi \cos{\left(2 t \right)}}{40} - \frac{\pi \cos{\left(10 t \right)}}{40}\right)d t}}}}{2}$$

Sabit katsayı kuralı $$$\int c f{\left(t \right)}\, dt = c \int f{\left(t \right)}\, dt$$$'i $$$c=\frac{1}{2}$$$ ve $$$f{\left(t \right)} = \frac{\pi \cos{\left(2 t \right)}}{20} - \frac{\pi \cos{\left(10 t \right)}}{20}$$$ ile uygula:

$$- \frac{\pi \sin{\left(2 t \right)}}{160} + \frac{\pi \sin{\left(6 t \right)}}{480} + \frac{{\color{red}{\int{\left(\frac{\pi \cos{\left(2 t \right)}}{40} - \frac{\pi \cos{\left(10 t \right)}}{40}\right)d t}}}}{2} = - \frac{\pi \sin{\left(2 t \right)}}{160} + \frac{\pi \sin{\left(6 t \right)}}{480} + \frac{{\color{red}{\left(\frac{\int{\left(\frac{\pi \cos{\left(2 t \right)}}{20} - \frac{\pi \cos{\left(10 t \right)}}{20}\right)d t}}{2}\right)}}}{2}$$

Her terimin integralini alın:

$$- \frac{\pi \sin{\left(2 t \right)}}{160} + \frac{\pi \sin{\left(6 t \right)}}{480} + \frac{{\color{red}{\int{\left(\frac{\pi \cos{\left(2 t \right)}}{20} - \frac{\pi \cos{\left(10 t \right)}}{20}\right)d t}}}}{4} = - \frac{\pi \sin{\left(2 t \right)}}{160} + \frac{\pi \sin{\left(6 t \right)}}{480} + \frac{{\color{red}{\left(\int{\frac{\pi \cos{\left(2 t \right)}}{20} d t} - \int{\frac{\pi \cos{\left(10 t \right)}}{20} d t}\right)}}}{4}$$

Sabit katsayı kuralı $$$\int c f{\left(t \right)}\, dt = c \int f{\left(t \right)}\, dt$$$'i $$$c=\frac{\pi}{20}$$$ ve $$$f{\left(t \right)} = \cos{\left(10 t \right)}$$$ ile uygula:

$$- \frac{\pi \sin{\left(2 t \right)}}{160} + \frac{\pi \sin{\left(6 t \right)}}{480} + \frac{\int{\frac{\pi \cos{\left(2 t \right)}}{20} d t}}{4} - \frac{{\color{red}{\int{\frac{\pi \cos{\left(10 t \right)}}{20} d t}}}}{4} = - \frac{\pi \sin{\left(2 t \right)}}{160} + \frac{\pi \sin{\left(6 t \right)}}{480} + \frac{\int{\frac{\pi \cos{\left(2 t \right)}}{20} d t}}{4} - \frac{{\color{red}{\left(\frac{\pi \int{\cos{\left(10 t \right)} d t}}{20}\right)}}}{4}$$

$$$u=10 t$$$ olsun.

Böylece $$$du=\left(10 t\right)^{\prime }dt = 10 dt$$$ (adımlar » görülebilir) ve $$$dt = \frac{du}{10}$$$ elde ederiz.

İntegral şu şekilde yeniden yazılabilir:

$$- \frac{\pi \sin{\left(2 t \right)}}{160} + \frac{\pi \sin{\left(6 t \right)}}{480} + \frac{\int{\frac{\pi \cos{\left(2 t \right)}}{20} d t}}{4} - \frac{\pi {\color{red}{\int{\cos{\left(10 t \right)} d t}}}}{80} = - \frac{\pi \sin{\left(2 t \right)}}{160} + \frac{\pi \sin{\left(6 t \right)}}{480} + \frac{\int{\frac{\pi \cos{\left(2 t \right)}}{20} d t}}{4} - \frac{\pi {\color{red}{\int{\frac{\cos{\left(u \right)}}{10} d u}}}}{80}$$

Sabit katsayı kuralı $$$\int c f{\left(u \right)}\, du = c \int f{\left(u \right)}\, du$$$'i $$$c=\frac{1}{10}$$$ ve $$$f{\left(u \right)} = \cos{\left(u \right)}$$$ ile uygula:

$$- \frac{\pi \sin{\left(2 t \right)}}{160} + \frac{\pi \sin{\left(6 t \right)}}{480} + \frac{\int{\frac{\pi \cos{\left(2 t \right)}}{20} d t}}{4} - \frac{\pi {\color{red}{\int{\frac{\cos{\left(u \right)}}{10} d u}}}}{80} = - \frac{\pi \sin{\left(2 t \right)}}{160} + \frac{\pi \sin{\left(6 t \right)}}{480} + \frac{\int{\frac{\pi \cos{\left(2 t \right)}}{20} d t}}{4} - \frac{\pi {\color{red}{\left(\frac{\int{\cos{\left(u \right)} d u}}{10}\right)}}}{80}$$

Kosinüsün integrali $$$\int{\cos{\left(u \right)} d u} = \sin{\left(u \right)}$$$:

$$- \frac{\pi \sin{\left(2 t \right)}}{160} + \frac{\pi \sin{\left(6 t \right)}}{480} + \frac{\int{\frac{\pi \cos{\left(2 t \right)}}{20} d t}}{4} - \frac{\pi {\color{red}{\int{\cos{\left(u \right)} d u}}}}{800} = - \frac{\pi \sin{\left(2 t \right)}}{160} + \frac{\pi \sin{\left(6 t \right)}}{480} + \frac{\int{\frac{\pi \cos{\left(2 t \right)}}{20} d t}}{4} - \frac{\pi {\color{red}{\sin{\left(u \right)}}}}{800}$$

Hatırlayın ki $$$u=10 t$$$:

$$- \frac{\pi \sin{\left(2 t \right)}}{160} + \frac{\pi \sin{\left(6 t \right)}}{480} + \frac{\int{\frac{\pi \cos{\left(2 t \right)}}{20} d t}}{4} - \frac{\pi \sin{\left({\color{red}{u}} \right)}}{800} = - \frac{\pi \sin{\left(2 t \right)}}{160} + \frac{\pi \sin{\left(6 t \right)}}{480} + \frac{\int{\frac{\pi \cos{\left(2 t \right)}}{20} d t}}{4} - \frac{\pi \sin{\left({\color{red}{\left(10 t\right)}} \right)}}{800}$$

Sabit katsayı kuralı $$$\int c f{\left(t \right)}\, dt = c \int f{\left(t \right)}\, dt$$$'i $$$c=\frac{\pi}{20}$$$ ve $$$f{\left(t \right)} = \cos{\left(2 t \right)}$$$ ile uygula:

$$- \frac{\pi \sin{\left(2 t \right)}}{160} + \frac{\pi \sin{\left(6 t \right)}}{480} - \frac{\pi \sin{\left(10 t \right)}}{800} + \frac{{\color{red}{\int{\frac{\pi \cos{\left(2 t \right)}}{20} d t}}}}{4} = - \frac{\pi \sin{\left(2 t \right)}}{160} + \frac{\pi \sin{\left(6 t \right)}}{480} - \frac{\pi \sin{\left(10 t \right)}}{800} + \frac{{\color{red}{\left(\frac{\pi \int{\cos{\left(2 t \right)} d t}}{20}\right)}}}{4}$$

İntegral $$$\int{\cos{\left(2 t \right)} d t}$$$ zaten hesaplandı:

$$\int{\cos{\left(2 t \right)} d t} = \frac{\sin{\left(2 t \right)}}{2}$$

Dolayısıyla,

$$- \frac{\pi \sin{\left(2 t \right)}}{160} + \frac{\pi \sin{\left(6 t \right)}}{480} - \frac{\pi \sin{\left(10 t \right)}}{800} + \frac{\pi {\color{red}{\int{\cos{\left(2 t \right)} d t}}}}{80} = - \frac{\pi \sin{\left(2 t \right)}}{160} + \frac{\pi \sin{\left(6 t \right)}}{480} - \frac{\pi \sin{\left(10 t \right)}}{800} + \frac{\pi {\color{red}{\left(\frac{\sin{\left(2 t \right)}}{2}\right)}}}{80}$$

Dolayısıyla,

$$\int{\frac{\pi \sin{\left(2 t \right)} \sin{\left(4 t \right)} \cos{\left(4 t \right)}}{20} d t} = \frac{\pi \sin{\left(6 t \right)}}{480} - \frac{\pi \sin{\left(10 t \right)}}{800}$$

Sadeleştirin:

$$\int{\frac{\pi \sin{\left(2 t \right)} \sin{\left(4 t \right)} \cos{\left(4 t \right)}}{20} d t} = \frac{\pi \left(5 \sin{\left(6 t \right)} - 3 \sin{\left(10 t \right)}\right)}{2400}$$

İntegrasyon sabitini ekleyin:

$$\int{\frac{\pi \sin{\left(2 t \right)} \sin{\left(4 t \right)} \cos{\left(4 t \right)}}{20} d t} = \frac{\pi \left(5 \sin{\left(6 t \right)} - 3 \sin{\left(10 t \right)}\right)}{2400}+C$$

$$$\int \frac{\pi \sin{\left(2 t \right)} \sin{\left(4 t \right)} \cos{\left(4 t \right)}}{20}\, dt = \frac{\pi \left(5 \sin{\left(6 t \right)} - 3 \sin{\left(10 t \right)}\right)}{2400} + C$$$A