Integral of $$$d \delta \cos^{4}{\left(\delta \right)}$$$ with respect to $$$d$$$

Find $$$\int d \delta \cos^{4}{\left(\delta \right)}\, dd$$$.

Apply the constant multiple rule $$$\int c f{\left(d \right)}\, dd = c \int f{\left(d \right)}\, dd$$$ with $$$c=\delta \cos^{4}{\left(\delta \right)}$$$ and $$$f{\left(d \right)} = d$$$:

$${\color{red}{\int{d \delta \cos^{4}{\left(\delta \right)} d d}}} = {\color{red}{\delta \cos^{4}{\left(\delta \right)} \int{d d d}}}$$

Apply the power rule $$$\int d^{n}\, dd = \frac{d^{n + 1}}{n + 1}$$$ $$$\left(n \neq -1 \right)$$$ with $$$n=1$$$:

$$\delta \cos^{4}{\left(\delta \right)} {\color{red}{\int{d d d}}}=\delta \cos^{4}{\left(\delta \right)} {\color{red}{\frac{d^{1 + 1}}{1 + 1}}}=\delta \cos^{4}{\left(\delta \right)} {\color{red}{\left(\frac{d^{2}}{2}\right)}}$$

Therefore,

$$\int{d \delta \cos^{4}{\left(\delta \right)} d d} = \frac{d^{2} \delta \cos^{4}{\left(\delta \right)}}{2}$$

Add the constant of integration:

$$\int{d \delta \cos^{4}{\left(\delta \right)} d d} = \frac{d^{2} \delta \cos^{4}{\left(\delta \right)}}{2}+C$$

$$$\int d \delta \cos^{4}{\left(\delta \right)}\, dd = \frac{d^{2} \delta \cos^{4}{\left(\delta \right)}}{2} + C$$$A