Funktion $$$\frac{4}{\sqrt{1 - x^{2}}}$$$ integraali

Määritä $$$\int \frac{4}{\sqrt{1 - x^{2}}}\, dx$$$.

Sovella vakiokertoimen sääntöä $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$ käyttäen $$$c=4$$$ ja $$$f{\left(x \right)} = \frac{1}{\sqrt{1 - x^{2}}}$$$:

$${\color{red}{\int{\frac{4}{\sqrt{1 - x^{2}}} d x}}} = {\color{red}{\left(4 \int{\frac{1}{\sqrt{1 - x^{2}}} d x}\right)}}$$

Funktion $$$\frac{1}{\sqrt{1 - x^{2}}}$$$ integraali on $$$\int{\frac{1}{\sqrt{1 - x^{2}}} d x} = \operatorname{asin}{\left(x \right)}$$$:

$$4 {\color{red}{\int{\frac{1}{\sqrt{1 - x^{2}}} d x}}} = 4 {\color{red}{\operatorname{asin}{\left(x \right)}}}$$

Näin ollen,

$$\int{\frac{4}{\sqrt{1 - x^{2}}} d x} = 4 \operatorname{asin}{\left(x \right)}$$

Lisää integrointivakio:

$$\int{\frac{4}{\sqrt{1 - x^{2}}} d x} = 4 \operatorname{asin}{\left(x \right)}+C$$

$$$\int \frac{4}{\sqrt{1 - x^{2}}}\, dx = 4 \operatorname{asin}{\left(x \right)} + C$$$A