Funktion $$$\cos^{5}{\left(x \right)}$$$ integraali

Määritä $$$\int \cos^{5}{\left(x \right)}\, dx$$$.

Irrota yksi kosini ja kirjoita kaikki muu sinin termeinä, käyttäen kaavaa $$$\cos^2\left(\alpha \right)=-\sin^2\left(\alpha \right)+1$$$ siten, että $$$\alpha=x$$$:

$${\color{red}{\int{\cos^{5}{\left(x \right)} d x}}} = {\color{red}{\int{\left(1 - \sin^{2}{\left(x \right)}\right)^{2} \cos{\left(x \right)} d x}}}$$

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

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

Siis,

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

Expand the expression:

$${\color{red}{\int{\left(1 - u^{2}\right)^{2} d u}}} = {\color{red}{\int{\left(u^{4} - 2 u^{2} + 1\right)d u}}}$$

Integroi termi kerrallaan:

$${\color{red}{\int{\left(u^{4} - 2 u^{2} + 1\right)d u}}} = {\color{red}{\left(\int{1 d u} - \int{2 u^{2} d u} + \int{u^{4} d u}\right)}}$$

Sovella vakiosääntöä $$$\int c\, du = c u$$$ käyttäen $$$c=1$$$:

$$- \int{2 u^{2} d u} + \int{u^{4} d u} + {\color{red}{\int{1 d u}}} = - \int{2 u^{2} d u} + \int{u^{4} d u} + {\color{red}{u}}$$

Sovella potenssisääntöä $$$\int u^{n}\, du = \frac{u^{n + 1}}{n + 1}$$$ $$$\left(n \neq -1 \right)$$$ käyttäen $$$n=4$$$:

$$u - \int{2 u^{2} d u} + {\color{red}{\int{u^{4} d u}}}=u - \int{2 u^{2} d u} + {\color{red}{\frac{u^{1 + 4}}{1 + 4}}}=u - \int{2 u^{2} d u} + {\color{red}{\left(\frac{u^{5}}{5}\right)}}$$

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

$$\frac{u^{5}}{5} + u - {\color{red}{\int{2 u^{2} d u}}} = \frac{u^{5}}{5} + u - {\color{red}{\left(2 \int{u^{2} d u}\right)}}$$

Sovella potenssisääntöä $$$\int u^{n}\, du = \frac{u^{n + 1}}{n + 1}$$$ $$$\left(n \neq -1 \right)$$$ käyttäen $$$n=2$$$:

$$\frac{u^{5}}{5} + u - 2 {\color{red}{\int{u^{2} d u}}}=\frac{u^{5}}{5} + u - 2 {\color{red}{\frac{u^{1 + 2}}{1 + 2}}}=\frac{u^{5}}{5} + u - 2 {\color{red}{\left(\frac{u^{3}}{3}\right)}}$$

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

$${\color{red}{u}} - \frac{2 {\color{red}{u}}^{3}}{3} + \frac{{\color{red}{u}}^{5}}{5} = {\color{red}{\sin{\left(x \right)}}} - \frac{2 {\color{red}{\sin{\left(x \right)}}}^{3}}{3} + \frac{{\color{red}{\sin{\left(x \right)}}}^{5}}{5}$$

Näin ollen,

$$\int{\cos^{5}{\left(x \right)} d x} = \frac{\sin^{5}{\left(x \right)}}{5} - \frac{2 \sin^{3}{\left(x \right)}}{3} + \sin{\left(x \right)}$$

Lisää integrointivakio:

$$\int{\cos^{5}{\left(x \right)} d x} = \frac{\sin^{5}{\left(x \right)}}{5} - \frac{2 \sin^{3}{\left(x \right)}}{3} + \sin{\left(x \right)}+C$$

$$$\int \cos^{5}{\left(x \right)}\, dx = \left(\frac{\sin^{5}{\left(x \right)}}{5} - \frac{2 \sin^{3}{\left(x \right)}}{3} + \sin{\left(x \right)}\right) + C$$$A