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The value of \( \int_{-\Pi / 4}^{\Pi / 4} \sin ^{103} x \cdot \cos ^{101} x \mathrm{~d} \mathrm{x} \) is
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Given that $\int_{-\frac{\pi}{4}}^{\frac{\pi}{4}} \sin ^{103} x \cdot \cos ^{101} x d x$
Let $f(x)=\sin ^{103} x \cos ^{101} x$
Then $f(-x)=\sin ^{103}(-x) \cos ^{101}(-x)$
$=-\sin ^{103} x \cos ^{101} x=-f(x)$
So, it is an odd function.
Since, $\int_{-a}^{a} f(x) d x=0$
Therefore $\int_{-\frac{\pi}{4}}^{\frac{\pi}{4}} \sin ^{103} x \cdot \cos ^{101} x d x=0$
Let $f(x)=\sin ^{103} x \cos ^{101} x$
Then $f(-x)=\sin ^{103}(-x) \cos ^{101}(-x)$
$=-\sin ^{103} x \cos ^{101} x=-f(x)$
So, it is an odd function.
Since, $\int_{-a}^{a} f(x) d x=0$
Therefore $\int_{-\frac{\pi}{4}}^{\frac{\pi}{4}} \sin ^{103} x \cdot \cos ^{101} x d x=0$
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