sum_{n=1}^{10} int_{-2n-1}^{-2n} sin^{27} x , | vedclass

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(D) माना $I = \sum_{n=1}^{10} \int_{-2n-1}^{-2n} \sin^{27} x \, dx + \sum_{n=1}^{10} \int_{2n}^{2n+1} \sin^{27} x \, dx$. प्रथम समाकलन पद पर विचार करे

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(D) माना $I = \sum_{n=1}^{10} \int_{-2n-1}^{-2n} \sin^{27} x \, dx + \sum_{n=1}^{10} \int_{2n}^{2n+1} \sin^{27} x \, dx$. प्रथम समाकलन पद पर विचार करें: $J_n = \int_{-2n-1}^{-2n} \sin^{27} x \, dx$. माना $x = -t$, तो $dx = -dt$. जब $x = -2n-1$, तब $t = 2n+1$. जब $x = -2n$, तब $t = 2n$. अतः, $J_n = \int_{2n+1}^{2n} \sin^{27}(-t) (-dt) = \int_{2n}^{2n+1} -\sin^{27} t \, dt = -\int_{2n}^{2n+1} \sin^{27} t \, dt$. इस मान को योग में प्रतिस्थापित करने पर, हमें प्राप्त होता है: $I = \sum_{n=1}^{10} (-\int_{2n}^{2n+1} \sin^{27} x \, dx) + \sum_{n=1}^{10} \int_{2n}^{2n+1} \sin^{27} x \, dx$. $I = -\sum_{n=1}^{10} \int_{2n}^{2n+1} \sin^{27} x \, dx + \sum_{n=1}^{10} \int_{2n}^{2n+1} \sin^{27} x \, dx = 0$.

$\sum_{n=1}^{10} \int_{-2n-1}^{-2n} \sin^{27} x \, dx + \sum_{n=1}^{10} \int_{2n}^{2n+1} \sin^{27} x \, dx$ का मान ज्ञात कीजिए।

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