\(\mathop \smallint \limits_0^{4{\rm{\pi }}} \left| {\cos {\rm{x}}} \right|{\rm{dx}}\) किसके बराबर है?

धारणा:

निश्चित समाकल के लिए गुण

• \({\rm{If\;f}}\left( {2{\rm{a}} - {\rm{x}}} \right) = {\rm{f}}\left( {\rm{x}} \right){\rm{\;then}}\mathop \smallint \nolimits_0^{2{\rm{a}}} {\rm{f}}\left( {\rm{x}} \right){\rm{dx}} = 2\mathop \smallint \nolimits_0^{\rm{a}} {\rm{f}}\left( {\rm{x}} \right){\rm{dx}}\)
• यदि f(x) अवधि T के साथ आवधिक फलन है तो \(\mathop \smallint \nolimits_0^{{\rm{nT}}} {\rm{f}}\left( {\rm{x}} \right){\rm{dx}} = {\rm{n}}\mathop \smallint \nolimits_0^{\rm{T}} {\rm{f}}\left( {\rm{x}} \right){\rm{dx}},{\rm{\;n}} \in {\rm{Z}}\)

गणना:

\({\rm{Let\;I\;}} = \;\mathop \smallint \limits_0^{4{\rm{\pi }}} \left| {\cos {\rm{x}}} \right|{\rm{dx}}\)

|cos x| की अवधि π है

\({\rm{I\;}} = {\rm{\;}}4\mathop \smallint \limits_0^{\rm{\pi }} \left| {\cos {\rm{x}}} \right|{\rm{dx}}\)

\(\left| {\cos x} \right| = \left\{ {\begin{array}{*{20}{c}} {\cos {\rm{x}},\;\;0 \le x < \frac{{\rm{\pi }}}{2}}\\ { - \cos {\rm{x}},\;\;\frac{{\rm{\pi }}}{2} \le x < \pi } \end{array}} \right.\)

\({\rm{I\;}} = {\rm{\;}}4\left[ {\mathop \smallint \limits_0^{\frac{{\rm{\pi }}}{2}} \cos {\rm{xdx}} + \mathop \smallint \limits_{\frac{{\rm{\pi }}}{2}}^{\rm{\pi }} \left( { - \cos {\rm{x}}} \right){\rm{dx}}} \right]\)

\( = {\rm{\;}}4{\rm{\;}}\left[ {\sin x} \right]_0^{\frac{{\rm{\pi }}}{2}} - 4\left[ {\sin x} \right]_{\frac{{\rm{\pi }}}{2}}^{\rm{\pi }}\)

\(= {\rm{\;}}4{\rm{\;}}\left( {\sin \frac{{\rm{\pi }}}{2} - \;\sin 0} \right) - 4{\rm{\;}}\left( {\sin {\rm{\pi }} - \;\sin \frac{{\rm{\pi }}}{2}} \right)\)

= 4(1 – 0) – 4(0 – 1)

= 4 + 4 = 8