Q.For any value of $n\in\mathbb{Z}$, $\displaystyle\int_0^\pi e^{\cos^2x}\cos^3\big[(2n+1)x\big]\,dx$ is

Twelve working properties, all provable from the Second Fundamental Theorem, that let a definite integral be simplified — often to $0$ or to a much easier integral — without direct evaluation. Throughout, $f,g$ are continuous on the relevant interval and $\alpha,\beta$ are constants.

1. Dummy-variable invariance: $\displaystyle\int_a^b f(x)\,dx=\int_a^b f(u)\,du$ — the integration variable's name never matters.
2. Limit reversal: $\displaystyle\int_b^a f(x)\,dx=-\int_a^b f(x)\,dx$.
3. Additivity: $\displaystyle\int_a^b f(x)\,dx=\int_a^c f(x)\,dx+\int_c^b f(x)\,dx$ for $a<c<b$.
4. Linearity: $\displaystyle\int_a^b[\alpha f(x)+\beta g(x)]\,dx=\alpha\int_a^b f(x)\,dx+\beta\int_a^b g(x)\,dx$.
5. Substitution $x=g(u)$: $\displaystyle\int_a^b f(x)\,dx=\int_c^d f(g(u))\,g'(u)\,du$ where $g(c)=a,\ g(d)=b$ — the tool for evaluating by substitution.
6. The $a+b-x$ trick: $\displaystyle\int_a^b f(x)\,dx=\int_a^b f(a+b-x)\,dx$; taking $a=0$ gives the very common special case $\int_0^a f(x)\,dx=\int_0^a f(a-x)\,dx$.
7. The $2a-x$ split: $\displaystyle\int_0^{2a} f(x)\,dx=\int_0^a\big[f(x)+f(2a-x)\big]\,dx$.
8. Even-function shortcut: if $f(-x)=f(x)$ (even), then $\displaystyle\int_{-a}^a f(x)\,dx=2\int_0^a f(x)\,dx$.
9. Odd-function shortcut: if $f(-x)=-f(x)$ (odd), then $\displaystyle\int_{-a}^a f(x)\,dx=0$.
10. Half-period doubling: if $f(2a-x)=f(x)$, then $\displaystyle\int_0^{2a} f(x)\,dx=2\int_0^a f(x)\,dx$ (follows from Property 7).
11. Half-period cancellation: if $f(2a-x)=-f(x)$, then $\displaystyle\int_0^{2a} f(x)\,dx=0$ (also from Property 7).
12. The $x\,f(x)$ symmetry trick: if $f(a-x)=f(x)$, then $\displaystyle\int_0^a x\,f(x)\,dx=\dfrac{a}{2}\int_0^a f(x)\,dx$ — removes the extra factor of $x$ from the integrand.