Integral of $$$5 e^{5 s} \sin{\left(e^{5 s} \right)}$$$

Find $$$\int 5 e^{5 s} \sin{\left(e^{5 s} \right)}\, ds$$$.

Apply the constant multiple rule $$$\int c f{\left(s \right)}\, ds = c \int f{\left(s \right)}\, ds$$$ with $$$c=5$$$ and $$$f{\left(s \right)} = e^{5 s} \sin{\left(e^{5 s} \right)}$$$:

$${\color{red}{\int{5 e^{5 s} \sin{\left(e^{5 s} \right)} d s}}} = {\color{red}{\left(5 \int{e^{5 s} \sin{\left(e^{5 s} \right)} d s}\right)}}$$

Let $$$u=5 s$$$.

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

Therefore,

$$5 {\color{red}{\int{e^{5 s} \sin{\left(e^{5 s} \right)} d s}}} = 5 {\color{red}{\int{\frac{e^{u} \sin{\left(e^{u} \right)}}{5} d u}}}$$

Apply the constant multiple rule $$$\int c f{\left(u \right)}\, du = c \int f{\left(u \right)}\, du$$$ with $$$c=\frac{1}{5}$$$ and $$$f{\left(u \right)} = e^{u} \sin{\left(e^{u} \right)}$$$:

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

Let $$$v=e^{u}$$$.

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

Therefore,

$${\color{red}{\int{e^{u} \sin{\left(e^{u} \right)} d u}}} = {\color{red}{\int{\sin{\left(v \right)} d v}}}$$

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

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

Recall that $$$v=e^{u}$$$:

$$- \cos{\left({\color{red}{v}} \right)} = - \cos{\left({\color{red}{e^{u}}} \right)}$$

Recall that $$$u=5 s$$$:

$$- \cos{\left(e^{{\color{red}{u}}} \right)} = - \cos{\left(e^{{\color{red}{\left(5 s\right)}}} \right)}$$

Therefore,

$$\int{5 e^{5 s} \sin{\left(e^{5 s} \right)} d s} = - \cos{\left(e^{5 s} \right)}$$

Add the constant of integration:

$$\int{5 e^{5 s} \sin{\left(e^{5 s} \right)} d s} = - \cos{\left(e^{5 s} \right)}+C$$

$$$\int 5 e^{5 s} \sin{\left(e^{5 s} \right)}\, ds = - \cos{\left(e^{5 s} \right)} + C$$$A