Nếu \(\int\limits_0^a {\left( {\cos x + \sin x} \right)dx}  = 0\left( {0 < a < 2\pi } \right)\) thì giá trị của \(a\) là:

Ta có:

 \(\begin{array}{l}\int\limits_0^a {\left( {\cos x + \sin x} \right)dx}  = 0 \Leftrightarrow \left. {\sin x} \right|_0^a - \left. {\cos x} \right|_0^a = 0 \Leftrightarrow \sin a - \cos a + 1 = 0\\ \Leftrightarrow \sqrt 2 \sin \left( {a - \dfrac{\pi }{4}} \right) =  - 1 \Leftrightarrow \sin \left( {a - \dfrac{\pi }{4}} \right) =  - \dfrac{1}{{\sqrt 2 }} \Leftrightarrow \left[ \begin{array}{l}a - \dfrac{\pi }{4} =  - \dfrac{\pi }{4} + k2\pi \\a - \dfrac{\pi }{4} = \dfrac{{5\pi }}{4} + k2\pi \end{array} \right. \Leftrightarrow a = \dfrac{{3\pi }}{2}\left( {0 < a < 2\pi } \right)\end{array}\)