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Integration problems, help!

Prove that,\[\int_{0}^{\infty}\dfrac{1}{x^3}\sin^3\left(x-\dfrac{1}{x} \right)^5\, dx=0\] and\[\int_{-1}^{0}\dfrac{x^2+2x}{\ln{(x+1)}} dx=\ln{3}\].I have no idea how to solve these problems!

Note by Adarsh Kumar
1 year, 9 months ago

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Comment deleted Jun 28, 2015

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@Samarpit Swain Sir, Ur Solution is wrong. Please check it again. There are a lot of mistakes. Rajdeep Dhingra · 1 year, 9 months ago

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@Samarpit Swain How did you get from the second last step to the last step,or more precisely how did you integrate those functions? Adarsh Kumar · 1 year, 9 months ago

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@Adarsh Kumar @Rajdeep Dhingra The first Integral is really hard. Also, it is not equal to \(0.\) The Integral is equivalent to \[-\frac15 \int_0^\infty \sin^3(u) u^{-3/5} du \approx {-0.1535} \] Anyway, to be able to compute the original Integral, you need to know some conditions for which an Integral becomes Riemann Improper Integrable. I can try to proceed if you want, but this is really miles and miles above our level. Ishan Dasgupta Samarendra · 1 year, 9 months ago

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@Adarsh Kumar Seems you've got some excellent answers. I would have done the second the same way as Rajdeep. Ishan Dasgupta Samarendra · 1 year, 9 months ago

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@Ishan Dasgupta Samarendra Sir the 1st one ? @Samarpit Swain solution is wrong. Rajdeep Dhingra · 1 year, 9 months ago

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@Rajdeep Dhingra By the way really nice solution! Ishan Dasgupta Samarendra · 1 year, 9 months ago

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@Ishan Dasgupta Samarendra Thank You sir Rajdeep Dhingra · 1 year, 9 months ago

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@Rajdeep Dhingra Come on Rajdeep! I'm not Sir!\[\]I'll have a go at it in half an hour. Ishan Dasgupta Samarendra · 1 year, 9 months ago

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@Ishan Dasgupta Samarendra Yes sir U are "sir" for me. Rajdeep Dhingra · 1 year, 9 months ago

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Comment deleted Jun 29, 2015

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@Rajdeep Dhingra U made terrible mistakes Kyle Finch · 1 year, 9 months ago

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@Kyle Finch Sir , I don't see any. Can U please tell ? Rajdeep Dhingra · 1 year, 9 months ago

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@Rajdeep Dhingra oh sorry, on looking closely it shows \(\frac{1}{a+1}\) otherwise i thought u had written \(\frac{1}{a-1}\) Kyle Finch · 1 year, 9 months ago

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@Rajdeep Dhingra Wow!Thanx for your solution!You are a genius! Adarsh Kumar · 1 year, 9 months ago

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@Adarsh Kumar Thanks ! I am removing this solution.(as you have seen it) Rajdeep Dhingra · 1 year, 9 months ago

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Hint about second integral, take x+1=t, and do it with the help of differentiation under integral. Ronak Agarwal · 1 year, 9 months ago

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@Ronak Agarwal Actually i tried that and I got,\[\int_{0}^{1}\dfrac{t^2-1}{\ln{t}}dt\],how to proceed now? Adarsh Kumar · 1 year, 9 months ago

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Hello Adarsh. I'm really sorry but I won't be able to help now; I'm truly exhausted after coaching class since 10 in the morning - and the same schedule is in place for tomorrow. I'll answer as soon as I can for sure. Ishan Dasgupta Samarendra · 1 year, 9 months ago

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@Ishan Dasgupta Samarendra No problem bhaiya!I understand! Adarsh Kumar · 1 year, 9 months ago

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