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 \[ \int_{2}^{5}f(x)\, dx=\int_{2}^{0}f(x)\, dx+\int_{0}^{5}f(x)\, dx=6+8=14 \]