Funktion $$$\frac{\sqrt{2} \sin{\left(2 x \right)}}{\cos^{3}{\left(x \right)}}$$$ integraali

Määritä $$$\int \frac{\sqrt{2} \sin{\left(2 x \right)}}{\cos^{3}{\left(x \right)}}\, dx$$$.

Kirjoita integroituva uudelleen:

$${\color{red}{\int{\frac{\sqrt{2} \sin{\left(2 x \right)}}{\cos^{3}{\left(x \right)}} d x}}} = {\color{red}{\int{\frac{2 \sqrt{2} \sin{\left(x \right)}}{\cos^{2}{\left(x \right)}} d x}}}$$

Sovella vakiokertoimen sääntöä $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$ käyttäen $$$c=2 \sqrt{2}$$$ ja $$$f{\left(x \right)} = \frac{\sin{\left(x \right)}}{\cos^{2}{\left(x \right)}}$$$:

$${\color{red}{\int{\frac{2 \sqrt{2} \sin{\left(x \right)}}{\cos^{2}{\left(x \right)}} d x}}} = {\color{red}{\left(2 \sqrt{2} \int{\frac{\sin{\left(x \right)}}{\cos^{2}{\left(x \right)}} d x}\right)}}$$

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 $$$\sin{\left(x \right)} dx = - du$$$.

Siis,

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

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

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

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

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

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

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

Näin ollen,

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

Lisää integrointivakio:

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

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