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Numerade Educator

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Problem 92 Hard Difficulty

If $ f $ is continuous on $ [0, \pi] $, use the substitution $ u = \pi - x $ to show that
$$ \int^{\pi}_0 x f(\sin x) \,dx = \frac{\pi}{2} \int^{\pi}_0 f(\sin x) \,dx $$

Answer

$\int_{0}^{\pi} x f(\sin x) d x=\frac{\pi}{2} \int_{0}^{\pi} f(\sin x) d x$

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Video Transcript

in order to prove this, the first thing we can do is set you equal to pi minus acts, which means acts. Is pi minus you? Therefore, we know that detox is equivalent to negative, Do you? Which means now our limits of integration change to pie. Mine's here on the bottom, which is Pai Pai Mei spanned the top, which is years. And that was just flipping the limits of integration and not plugging in in terms of are you. So after a sign of you, do you again, we're just plugging in. And now, lastly, we know the last step is we have to substitute back our acts and essentially take out the use. And now we have off of sign of axe DX because we don't want you in our final answer.