Integraali $$$a^{2} \cos{\left(x \right)} - x^{2}$$$:stä muuttujan $$$x$$$ suhteen

Määritä $$$\int \left(a^{2} \cos{\left(x \right)} - x^{2}\right)\, dx$$$.

Integroi termi kerrallaan:

$${\color{red}{\int{\left(a^{2} \cos{\left(x \right)} - x^{2}\right)d x}}} = {\color{red}{\left(- \int{x^{2} d x} + \int{a^{2} \cos{\left(x \right)} d x}\right)}}$$

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

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

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

$$- \frac{x^{3}}{3} + {\color{red}{\int{a^{2} \cos{\left(x \right)} d x}}} = - \frac{x^{3}}{3} + {\color{red}{a^{2} \int{\cos{\left(x \right)} d x}}}$$

Kosinin integraali on $$$\int{\cos{\left(x \right)} d x} = \sin{\left(x \right)}$$$:

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

Näin ollen,

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

Lisää integrointivakio:

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

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