Integraali $$$4 t \sin{\left(c^{2} \right)}$$$:stä muuttujan $$$t$$$ suhteen

Määritä $$$\int 4 t \sin{\left(c^{2} \right)}\, dt$$$.

Sovella vakiokertoimen sääntöä $$$\int c f{\left(t \right)}\, dt = c \int f{\left(t \right)}\, dt$$$ käyttäen $$$c=4 \sin{\left(c^{2} \right)}$$$ ja $$$f{\left(t \right)} = t$$$:

$${\color{red}{\int{4 t \sin{\left(c^{2} \right)} d t}}} = {\color{red}{\left(4 \sin{\left(c^{2} \right)} \int{t d t}\right)}}$$

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

$$4 \sin{\left(c^{2} \right)} {\color{red}{\int{t d t}}}=4 \sin{\left(c^{2} \right)} {\color{red}{\frac{t^{1 + 1}}{1 + 1}}}=4 \sin{\left(c^{2} \right)} {\color{red}{\left(\frac{t^{2}}{2}\right)}}$$

Näin ollen,

$$\int{4 t \sin{\left(c^{2} \right)} d t} = 2 t^{2} \sin{\left(c^{2} \right)}$$

Lisää integrointivakio:

$$\int{4 t \sin{\left(c^{2} \right)} d t} = 2 t^{2} \sin{\left(c^{2} \right)}+C$$

$$$\int 4 t \sin{\left(c^{2} \right)}\, dt = 2 t^{2} \sin{\left(c^{2} \right)} + C$$$A