Integral of $$$\frac{5 x^{3} \sin{\left(\frac{3 x^{2}}{5} \right)}}{3}$$$

Find $$$\int \frac{5 x^{3} \sin{\left(\frac{3 x^{2}}{5} \right)}}{3}\, dx$$$.

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

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

The integral becomes

$${\color{red}{\int{\frac{5 x^{3} \sin{\left(\frac{3 x^{2}}{5} \right)}}{3} d x}}} = {\color{red}{\int{\frac{5 u \sin{\left(\frac{3 u}{5} \right)}}{6} d u}}}$$

Apply the constant multiple rule $$$\int c f{\left(u \right)}\, du = c \int f{\left(u \right)}\, du$$$ with $$$c=\frac{5}{6}$$$ and $$$f{\left(u \right)} = u \sin{\left(\frac{3 u}{5} \right)}$$$:

$${\color{red}{\int{\frac{5 u \sin{\left(\frac{3 u}{5} \right)}}{6} d u}}} = {\color{red}{\left(\frac{5 \int{u \sin{\left(\frac{3 u}{5} \right)} d u}}{6}\right)}}$$

For the integral $$$\int{u \sin{\left(\frac{3 u}{5} \right)} d u}$$$, use integration by parts $$$\int \operatorname{t} \operatorname{dv} = \operatorname{t}\operatorname{v} - \int \operatorname{v} \operatorname{dt}$$$.

Let $$$\operatorname{t}=u$$$ and $$$\operatorname{dv}=\sin{\left(\frac{3 u}{5} \right)} du$$$.

Then $$$\operatorname{dt}=\left(u\right)^{\prime }du=1 du$$$ (steps can be seen ») and $$$\operatorname{v}=\int{\sin{\left(\frac{3 u}{5} \right)} d u}=- \frac{5 \cos{\left(\frac{3 u}{5} \right)}}{3}$$$ (steps can be seen »).

The integral can be rewritten as

$$\frac{5 {\color{red}{\int{u \sin{\left(\frac{3 u}{5} \right)} d u}}}}{6}=\frac{5 {\color{red}{\left(u \cdot \left(- \frac{5 \cos{\left(\frac{3 u}{5} \right)}}{3}\right)-\int{\left(- \frac{5 \cos{\left(\frac{3 u}{5} \right)}}{3}\right) \cdot 1 d u}\right)}}}{6}=\frac{5 {\color{red}{\left(- \frac{5 u \cos{\left(\frac{3 u}{5} \right)}}{3} - \int{\left(- \frac{5 \cos{\left(\frac{3 u}{5} \right)}}{3}\right)d u}\right)}}}{6}$$

Apply the constant multiple rule $$$\int c f{\left(u \right)}\, du = c \int f{\left(u \right)}\, du$$$ with $$$c=- \frac{5}{3}$$$ and $$$f{\left(u \right)} = \cos{\left(\frac{3 u}{5} \right)}$$$:

$$- \frac{25 u \cos{\left(\frac{3 u}{5} \right)}}{18} - \frac{5 {\color{red}{\int{\left(- \frac{5 \cos{\left(\frac{3 u}{5} \right)}}{3}\right)d u}}}}{6} = - \frac{25 u \cos{\left(\frac{3 u}{5} \right)}}{18} - \frac{5 {\color{red}{\left(- \frac{5 \int{\cos{\left(\frac{3 u}{5} \right)} d u}}{3}\right)}}}{6}$$

Let $$$v=\frac{3 u}{5}$$$.

Then $$$dv=\left(\frac{3 u}{5}\right)^{\prime }du = \frac{3 du}{5}$$$ (steps can be seen »), and we have that $$$du = \frac{5 dv}{3}$$$.

The integral becomes

$$- \frac{25 u \cos{\left(\frac{3 u}{5} \right)}}{18} + \frac{25 {\color{red}{\int{\cos{\left(\frac{3 u}{5} \right)} d u}}}}{18} = - \frac{25 u \cos{\left(\frac{3 u}{5} \right)}}{18} + \frac{25 {\color{red}{\int{\frac{5 \cos{\left(v \right)}}{3} d v}}}}{18}$$

Apply the constant multiple rule $$$\int c f{\left(v \right)}\, dv = c \int f{\left(v \right)}\, dv$$$ with $$$c=\frac{5}{3}$$$ and $$$f{\left(v \right)} = \cos{\left(v \right)}$$$:

$$- \frac{25 u \cos{\left(\frac{3 u}{5} \right)}}{18} + \frac{25 {\color{red}{\int{\frac{5 \cos{\left(v \right)}}{3} d v}}}}{18} = - \frac{25 u \cos{\left(\frac{3 u}{5} \right)}}{18} + \frac{25 {\color{red}{\left(\frac{5 \int{\cos{\left(v \right)} d v}}{3}\right)}}}{18}$$

The integral of the cosine is $$$\int{\cos{\left(v \right)} d v} = \sin{\left(v \right)}$$$:

$$- \frac{25 u \cos{\left(\frac{3 u}{5} \right)}}{18} + \frac{125 {\color{red}{\int{\cos{\left(v \right)} d v}}}}{54} = - \frac{25 u \cos{\left(\frac{3 u}{5} \right)}}{18} + \frac{125 {\color{red}{\sin{\left(v \right)}}}}{54}$$

Recall that $$$v=\frac{3 u}{5}$$$:

$$- \frac{25 u \cos{\left(\frac{3 u}{5} \right)}}{18} + \frac{125 \sin{\left({\color{red}{v}} \right)}}{54} = - \frac{25 u \cos{\left(\frac{3 u}{5} \right)}}{18} + \frac{125 \sin{\left({\color{red}{\left(\frac{3 u}{5}\right)}} \right)}}{54}$$

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

$$\frac{125 \sin{\left(\frac{3 {\color{red}{u}}}{5} \right)}}{54} - \frac{25 {\color{red}{u}} \cos{\left(\frac{3 {\color{red}{u}}}{5} \right)}}{18} = \frac{125 \sin{\left(\frac{3 {\color{red}{x^{2}}}}{5} \right)}}{54} - \frac{25 {\color{red}{x^{2}}} \cos{\left(\frac{3 {\color{red}{x^{2}}}}{5} \right)}}{18}$$

Therefore,

$$\int{\frac{5 x^{3} \sin{\left(\frac{3 x^{2}}{5} \right)}}{3} d x} = - \frac{25 x^{2} \cos{\left(\frac{3 x^{2}}{5} \right)}}{18} + \frac{125 \sin{\left(\frac{3 x^{2}}{5} \right)}}{54}$$

Add the constant of integration:

$$\int{\frac{5 x^{3} \sin{\left(\frac{3 x^{2}}{5} \right)}}{3} d x} = - \frac{25 x^{2} \cos{\left(\frac{3 x^{2}}{5} \right)}}{18} + \frac{125 \sin{\left(\frac{3 x^{2}}{5} \right)}}{54}+C$$

$$$\int \frac{5 x^{3} \sin{\left(\frac{3 x^{2}}{5} \right)}}{3}\, dx = \left(- \frac{25 x^{2} \cos{\left(\frac{3 x^{2}}{5} \right)}}{18} + \frac{125 \sin{\left(\frac{3 x^{2}}{5} \right)}}{54}\right) + C$$$A