Consider the integral $$\int_{-\delta}^{\delta} cos(nx)\big (1-\frac{\lvert x \rvert}{\delta}\big)dx$$
Where $n$ and $\delta$ are constants. Are there any reasonably simple ways to solve this integral? Im not familiar with integrals of absolute value's and the method suggested by an online calculator was very long end tedious.
You should not apply the fundamental theorem of calculus directly. Instead, since the integrand is even, you have
$$ \int_{-\delta}^\delta \cos(nx)\big (1-\frac{\lvert x \rvert}{\delta}\big)dx =2\int_0^{\delta}\cos(nx)\big (1-\frac{\lvert x \rvert}{\delta}\big)dx =2\int_0^{\delta}\cos(nx)\big (1-\frac{x }{\delta}\big)dx $$
Now, by linearity, you can work out the following integrals: $$ \int \cos(nx)dx,\quad \int \cos(nx)\cdot x\,dx $$ and then combine the results.
In general, if you have an even function $f:[-a,a]\to\mathbb{R}$ (that is Riemann integrable), $$ \int_{-a}^a f(x)dx=2\int_0^af(x)dx $$