Integration by parts

[JPJ]

Hello there,


I'm attempting to solve int(t sin 2t dt) by parts instead of using u-substitution, by I stumbled upon the wrong answer, and I can't figure out what's my mistake.

Here's how it goes:

int(t sin 2t dt) =

= (-1/2)int[t*d(cos2t)] =

= (-1/2)[t*cos2t + int(cos2t*dt)] =

= (-1/2)t*cos2t - (1/2)int(cos2t*dt)=

= (-1/2)*t*cos2t - (1/4)sin2t.


The correct answer is (-1/2)*t*cos2t + (1/4)sin2t, but I can't seem to find the (-1) that would set my answer right.

ps.: I wanna solve it without any sort of substitution.

 

[tkhunny]

(-1/2)int[t*d(cos2t)] =

(-1/2)[t*cos2t + int(cos2t*dt)]

Right there. Make that "-" in the middle.

 

[JPJ]

Right there. Make that "-" in the middle.

Gosh, I'm stupid, I was using the property wrongly...sorry for bothering with my brain malfunctioning...thanks

 

[tkhunny]

Where did you learn this style of Integration by Parts? It's not normally this way in U.S. Mathematics Texts. It is my preference, but it's hard to find.

 

[JPJ]

Style? I was just trying to set the sentence to an integration by parts really, not using any heuristic style/approach or anything...where is this sort of "style" common? Europe? Asia? Eastern Europe?

 

[tkhunny]

I first met it when demonstrated by a Russian mathematician visiting the USA. This was MANY years ago, but it remains relatively absent from textbooks available in the U.S., at least from those I have seen.