Integriranje razvojem u red

Riješite integral $\int \sin x^{2} dx$ razvojem u red potencija, koristeći razvoj

$\displaystyle \sin x=\sum\limits_{n=0}^{ \infty }(-1)^{n} \frac{x^{2n+1}}{(2n+1)!}.$

Rješenje. Zadana podintegralna funkcija je $\sin x^{2}$ pa koristeći zadani razvoj sinusa dobivamo

$\displaystyle \sin x^{2}$	$\displaystyle ={}\sum\limits_{n=0}^{\infty }\left( -1\right) ^{n}\frac{\left( x... ...eft( 2n+1\right) !}=\left( -1\right) ^{n}\frac{ x^{4n+2}}{\left( 2n+1\right) !}$
	$\displaystyle =x^{2}-\frac{x^{6}}{3!}+\frac{x^{10}}{5!}-\frac{x^{14}}{7!}+\cdot... ...t( -1\right) ^{n-1}\frac{x^{2\left( 2n+1\right) }}{\left( 2n+1\right) !}+\cdots$

Odavde slijedi

$\displaystyle \int \sin x^{2} dx$	$\displaystyle =\int \left[ x^{2}-\frac{x^{6}}{3!}+\frac{x^{10}}{5!}- \frac{x^{1... ...left( -1\right) ^{n-1}\frac{x^{4n+2}}{\left( 2n+1\right) !}+\cdots \right] dx$
	$\displaystyle =\frac{x^{3}}{3}-\frac{x^{7}}{7\cdot 3!}+\frac{x^{11}}{11\cdot 5!... ...-1\right) ^{n-1}\frac{x^{4n+3}}{\left( 4n+3\right) \left( 2n+1\right) !}+\cdots$
	$\displaystyle ={}\sum\limits_{n=0}^{\infty }\left( -1\right) ^{n}\frac{x^{4n+3}}{\left( 4n+3\right) \left( 2n+1\right) !}+C.$