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

## Comments

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TopNewestThis 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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– Mayank Singh · 2 years, 1 month ago

nice!Log in to reply

This is called

Solution by Inspection.You can find several such techniques in JEE-Advanced Books. – Sandeep Bhardwaj · 2 years, 1 month agoLog in to reply

– Mayank Singh · 2 years, 1 month ago

can you recommend one, please?Log in to reply

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

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this question? I don't think the Gamma function is correct. – Ishan Dasgupta Samarendra · 2 years, 1 month ago

Sir, could you please look atLog in to reply

@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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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 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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– Mayank Singh · 2 years, 1 month ago

Thanks!Log in to reply

– Ishan Dasgupta Samarendra · 2 years, 1 month ago

Welcome!Log in to reply

@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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– Mayank Singh · 2 years, 1 month ago

Thanks againLog in to reply