If $f$ is an even function and $\int_{0}^{2}f(x)\, dx=-6$ and $\int_{-5}^{0}f(x)\, dx=8,$ find $\int_{2}^{5}f(x)\, dx.$

Solution $\int_{2}^{5}f(x)\, dx=\int_{2}^{0}f(x)\, dx+\int_{0}^{5}f(x)\, dx$

Now $\int_{2}^{0}f(x)\, dx=-\int_{0}^{2}f(x)\, dx=6.$

Since $f$ is even, $\int_{0}^{5}f(x)\, dx=$ $\int_{-5}^{0}f(x)\, dx$ $\,=8.$ Then $$\int_{2}^{5}f(x)\, dx=\int_{2}^{0}f(x)\, dx+\int_{0}^{5}f(x)\, dx=6+8=14$$