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Help me to integrate w/o knowing the function

Recently, I faced a problem to solve this integral

\(\int { xdy+ydx } \)

without any function \(y=f\left( x \right) \). A friend of mine helped me here. He told me to substitute \(xy=t\) so that \(xdy+ydx\) becomes \(dt\) and the integral gets evaluated to \(xy\)

Now I want to know more about this technique in calculus.

What this technique is known as?

Do we have a wiki about it here on Brilliant?

P.S.:I was totally awestruck when I solved this integral without knowing the function

Note by Mayank Singh
2 years, 1 month ago

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This is a simple Integration By parts question
\[\int{x dy} + \int{y dx} \\ \text{Integrating by parts we get} \\ x \int{dy} - \int{y dx} + \int {y dx} \\ = \boxed{xy} \]

Wiki Rajdeep Dhingra · 2 years, 1 month ago

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@Rajdeep Dhingra nice! Mayank Singh · 2 years, 1 month ago

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This is called Solution by Inspection. You can find several such techniques in JEE-Advanced Books. Sandeep Bhardwaj · 2 years, 1 month ago

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@Sandeep Bhardwaj can you recommend one, please? Mayank Singh · 2 years, 1 month ago

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@Mayank Singh Integral Calculus of Arihant Publication, that's a good one. It will help you master the whole integral calculus, but for that you will need to go through this book completely.

All the best ! Sandeep Bhardwaj · 2 years, 1 month ago

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@Sandeep Bhardwaj Sir, could you please look at this question? I don't think the Gamma function is correct. Ishan Dasgupta Samarendra · 2 years, 1 month ago

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@Sandeep Bhardwaj @Sandeep Bhardwaj Sir, please could you give me a link where I could buy the book? I couldn't find it on the Net. Ishan Dasgupta Samarendra · 2 years, 1 month ago

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@Sandeep Bhardwaj Thanks sir, I have the book and found out what I was looking for

I'll do that when we do calculus in our coaching Mayank Singh · 2 years, 1 month ago

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@Sandeep Bhardwaj sir, @Ishan Dasgupta Samarendra , @Azhaghu Roopesh M please help

Thanks! Mayank Singh · 2 years, 1 month ago

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@Mayank Singh @Mayank Singh Sorry for responding late, I just came on Brilliant. On a slightly higher level, you could try Problems in Calculus of One Variable. The thing is that you will have to go through this carefully. But according to me, it's an excellent book. It even has Physics based applications as examples and questions. Ishan Dasgupta Samarendra · 2 years, 1 month ago

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@Ishan Dasgupta Samarendra Thanks! Mayank Singh · 2 years, 1 month ago

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@Mayank Singh Welcome! Ishan Dasgupta Samarendra · 2 years, 1 month ago

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@Ishan Dasgupta Samarendra @Mayank Singh You could also try this site for the basics. Under the option 'Class Notes', look at Calculus II and III as well as Differential Equations. Ishan Dasgupta Samarendra · 2 years, 1 month ago

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@Ishan Dasgupta Samarendra Thanks again Mayank Singh · 2 years, 1 month ago

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