Integral of $$$y \sin{\left(y^{2} \right)}$$$

Find $$$\int y \sin{\left(y^{2} \right)}\, dy$$$.

Let $$$u=y^{2}$$$.

Then $$$du=\left(y^{2}\right)^{\prime }dy = 2 y dy$$$ (steps can be seen »), and we have that $$$y dy = \frac{du}{2}$$$.

Therefore,

$${\color{red}{\int{y \sin{\left(y^{2} \right)} d y}}} = {\color{red}{\int{\frac{\sin{\left(u \right)}}{2} d u}}}$$

Apply the constant multiple rule $$$\int c f{\left(u \right)}\, du = c \int f{\left(u \right)}\, du$$$ with $$$c=\frac{1}{2}$$$ and $$$f{\left(u \right)} = \sin{\left(u \right)}$$$:

$${\color{red}{\int{\frac{\sin{\left(u \right)}}{2} d u}}} = {\color{red}{\left(\frac{\int{\sin{\left(u \right)} d u}}{2}\right)}}$$

The integral of the sine is $$$\int{\sin{\left(u \right)} d u} = - \cos{\left(u \right)}$$$:

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

Recall that $$$u=y^{2}$$$:

$$- \frac{\cos{\left({\color{red}{u}} \right)}}{2} = - \frac{\cos{\left({\color{red}{y^{2}}} \right)}}{2}$$

Therefore,

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

Add the constant of integration:

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

$$$\int y \sin{\left(y^{2} \right)}\, dy = - \frac{\cos{\left(y^{2} \right)}}{2} + C$$$A