Integraali $$$\frac{int_{0}^{\pi} \sin^{3}{\left(x \right)} \cos^{8}{\left(x \right)}}{\pi}$$$:stä muuttujan $$$x$$$ suhteen

Määritä $$$\int \frac{int_{0}^{\pi} \sin^{3}{\left(x \right)} \cos^{8}{\left(x \right)}}{\pi}\, dx$$$.

Sovella vakiokertoimen sääntöä $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$ käyttäen $$$c=\frac{int_{0}^{\pi}}{\pi}$$$ ja $$$f{\left(x \right)} = \sin^{3}{\left(x \right)} \cos^{8}{\left(x \right)}$$$:

$${\color{red}{\int{\frac{int_{0}^{\pi} \sin^{3}{\left(x \right)} \cos^{8}{\left(x \right)}}{\pi} d x}}} = {\color{red}{\frac{int_{0}^{\pi} \int{\sin^{3}{\left(x \right)} \cos^{8}{\left(x \right)} d x}}{\pi}}}$$

Irrota yksi sini ja kirjoita kaikki muu kosinin termeinä käyttäen kaavaa $$$\sin^2\left(\alpha \right)=-\cos^2\left(\alpha \right)+1$$$, jossa $$$\alpha=x$$$:

$$\frac{int_{0}^{\pi} {\color{red}{\int{\sin^{3}{\left(x \right)} \cos^{8}{\left(x \right)} d x}}}}{\pi} = \frac{int_{0}^{\pi} {\color{red}{\int{\left(1 - \cos^{2}{\left(x \right)}\right) \sin{\left(x \right)} \cos^{8}{\left(x \right)} d x}}}}{\pi}$$

Olkoon $$$u=\cos{\left(x \right)}$$$.

Tällöin $$$du=\left(\cos{\left(x \right)}\right)^{\prime }dx = - \sin{\left(x \right)} dx$$$ (vaiheet ovat nähtävissä ») ja saamme, että $$$\sin{\left(x \right)} dx = - du$$$.

Näin ollen,

$$\frac{int_{0}^{\pi} {\color{red}{\int{\left(1 - \cos^{2}{\left(x \right)}\right) \sin{\left(x \right)} \cos^{8}{\left(x \right)} d x}}}}{\pi} = \frac{int_{0}^{\pi} {\color{red}{\int{\left(- u^{8} \left(1 - u^{2}\right)\right)d u}}}}{\pi}$$

Sovella vakiokertoimen sääntöä $$$\int c f{\left(u \right)}\, du = c \int f{\left(u \right)}\, du$$$ käyttäen $$$c=-1$$$ ja $$$f{\left(u \right)} = u^{8} \left(1 - u^{2}\right)$$$:

$$\frac{int_{0}^{\pi} {\color{red}{\int{\left(- u^{8} \left(1 - u^{2}\right)\right)d u}}}}{\pi} = \frac{int_{0}^{\pi} {\color{red}{\left(- \int{u^{8} \left(1 - u^{2}\right) d u}\right)}}}{\pi}$$

Expand the expression:

$$- \frac{int_{0}^{\pi} {\color{red}{\int{u^{8} \left(1 - u^{2}\right) d u}}}}{\pi} = - \frac{int_{0}^{\pi} {\color{red}{\int{\left(- u^{10} + u^{8}\right)d u}}}}{\pi}$$

Integroi termi kerrallaan:

$$- \frac{int_{0}^{\pi} {\color{red}{\int{\left(- u^{10} + u^{8}\right)d u}}}}{\pi} = - \frac{int_{0}^{\pi} {\color{red}{\left(\int{u^{8} d u} - \int{u^{10} d u}\right)}}}{\pi}$$

Sovella potenssisääntöä $$$\int u^{n}\, du = \frac{u^{n + 1}}{n + 1}$$$ $$$\left(n \neq -1 \right)$$$ käyttäen $$$n=8$$$:

$$- \frac{int_{0}^{\pi} \left(- \int{u^{10} d u} + {\color{red}{\int{u^{8} d u}}}\right)}{\pi}=- \frac{int_{0}^{\pi} \left(- \int{u^{10} d u} + {\color{red}{\frac{u^{1 + 8}}{1 + 8}}}\right)}{\pi}=- \frac{int_{0}^{\pi} \left(- \int{u^{10} d u} + {\color{red}{\left(\frac{u^{9}}{9}\right)}}\right)}{\pi}$$

Sovella potenssisääntöä $$$\int u^{n}\, du = \frac{u^{n + 1}}{n + 1}$$$ $$$\left(n \neq -1 \right)$$$ käyttäen $$$n=10$$$:

$$- \frac{int_{0}^{\pi} \left(\frac{u^{9}}{9} - {\color{red}{\int{u^{10} d u}}}\right)}{\pi}=- \frac{int_{0}^{\pi} \left(\frac{u^{9}}{9} - {\color{red}{\frac{u^{1 + 10}}{1 + 10}}}\right)}{\pi}=- \frac{int_{0}^{\pi} \left(\frac{u^{9}}{9} - {\color{red}{\left(\frac{u^{11}}{11}\right)}}\right)}{\pi}$$

Muista, että $$$u=\cos{\left(x \right)}$$$:

$$- \frac{int_{0}^{\pi} \left(\frac{{\color{red}{u}}^{9}}{9} - \frac{{\color{red}{u}}^{11}}{11}\right)}{\pi} = - \frac{int_{0}^{\pi} \left(\frac{{\color{red}{\cos{\left(x \right)}}}^{9}}{9} - \frac{{\color{red}{\cos{\left(x \right)}}}^{11}}{11}\right)}{\pi}$$

Näin ollen,

$$\int{\frac{int_{0}^{\pi} \sin^{3}{\left(x \right)} \cos^{8}{\left(x \right)}}{\pi} d x} = - \frac{int_{0}^{\pi} \left(- \frac{\cos^{11}{\left(x \right)}}{11} + \frac{\cos^{9}{\left(x \right)}}{9}\right)}{\pi}$$

Sievennä:

$$\int{\frac{int_{0}^{\pi} \sin^{3}{\left(x \right)} \cos^{8}{\left(x \right)}}{\pi} d x} = \frac{int_{0}^{\pi} \left(9 \cos^{2}{\left(x \right)} - 11\right) \cos^{9}{\left(x \right)}}{99 \pi}$$

Lisää integrointivakio:

$$\int{\frac{int_{0}^{\pi} \sin^{3}{\left(x \right)} \cos^{8}{\left(x \right)}}{\pi} d x} = \frac{int_{0}^{\pi} \left(9 \cos^{2}{\left(x \right)} - 11\right) \cos^{9}{\left(x \right)}}{99 \pi}+C$$

$$$\int \frac{int_{0}^{\pi} \sin^{3}{\left(x \right)} \cos^{8}{\left(x \right)}}{\pi}\, dx = \frac{int_{0}^{\pi} \left(9 \cos^{2}{\left(x \right)} - 11\right) \cos^{9}{\left(x \right)}}{99 \pi} + C$$$A