Integral of $$$\cos{\left(\frac{t}{a} \right)}$$$ with respect to $$$t$$$

Find $$$\int \cos{\left(\frac{t}{a} \right)}\, dt$$$.

Let $$$u=\frac{t}{a}$$$.

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

Therefore,

$${\color{red}{\int{\cos{\left(\frac{t}{a} \right)} d t}}} = {\color{red}{\int{a \cos{\left(u \right)} d u}}}$$

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

$${\color{red}{\int{a \cos{\left(u \right)} d u}}} = {\color{red}{a \int{\cos{\left(u \right)} d u}}}$$

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

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

Recall that $$$u=\frac{t}{a}$$$:

$$a \sin{\left({\color{red}{u}} \right)} = a \sin{\left({\color{red}{\frac{t}{a}}} \right)}$$

Therefore,

$$\int{\cos{\left(\frac{t}{a} \right)} d t} = a \sin{\left(\frac{t}{a} \right)}$$

Add the constant of integration:

$$\int{\cos{\left(\frac{t}{a} \right)} d t} = a \sin{\left(\frac{t}{a} \right)}+C$$

$$$\int \cos{\left(\frac{t}{a} \right)}\, dt = a \sin{\left(\frac{t}{a} \right)} + C$$$A