Question
Download Solution PDF\(\rm \int_{-\pi }^{\pi }cosx \ dx\) का मान ज्ञात कीजिए।
Answer (Detailed Solution Below)
Detailed Solution
Download Solution PDFअवधारणा:
फलन f(x) एक विषम फलन है अगर f(x) = - f(-x) और सम फलन है यदि f(x) = f(-x)।
- जब f(x) एक सम फलन है तो \(\rm \int_{-a}^{a}f(x)dx=2\int_{0}^{a}f(x)dx\)
- जब f(x) विषम फलन हो तो \(\rm \int_{-a}^{a}f(x)dx=0\)
दिया गया: \(\rm \int_{-\pi }^{\pi }cosx \ dx\)
माना कि f(x) = cos x
जैसा कि हम देख सकते हैं कि f(- x) = cos (- x) = cos x = f(x)।
तो, cos x एक सम फलन है।
जैसा कि हम जानते हैं कि, जब f(x) एक सम फलन है तो \(\rm \int_{-a}^{a}f(x)dx=2\int_{0}^{a}f(x)dx\)
\(\Rightarrow \rm \int_{-\pi }^{\pi }cosx \ dx=2\int_{0 }^{\pi }cosx \ dx=2(sin\pi-sin0)=0\)
इसलिए, सही विकल्प 1 है।
Last updated on Jun 30, 2025
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