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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 $$xy$$

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, 6 months ago

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

- 2 years, 6 months ago

nice!

- 2 years, 6 months ago

This is called Solution by Inspection. You can find several such techniques in JEE-Advanced Books.

- 2 years, 6 months ago

- 2 years, 6 months ago

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 !

- 2 years, 6 months ago

Sir, could you please look at this question? I don't think the Gamma function is correct.

- 2 years, 6 months ago

@Sandeep Bhardwaj Sir, please could you give me a link where I could buy the book? I couldn't find it on the Net.

- 2 years, 6 months ago

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

- 2 years, 6 months ago

Thanks!

- 2 years, 6 months ago

@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.

- 2 years, 6 months ago

Thanks!

- 2 years, 6 months ago

Welcome!

- 2 years, 6 months ago

@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.

- 2 years, 6 months ago

Thanks again

- 2 years, 6 months ago