Author Topic: HOW TO: Anti derivatives through derivatives. (Read 3570 times) Tweet Share

Collin Li · « **Reply #15 on:** November 06, 2008, 01:46:29 pm »

$\frac{d}{dx}\left[2x \sin{(3x)} \right] = 2\sin{(3x)} + 6x\cos{(3x)}$

Integrating both sides with respect to $x$ :

$\implies 2x \sin{(3x)} = \int 2\sin{(3x)}\, dx + \int 6x\cos{(3x)}\, dx$

$\implies \int 6x\cos{(3x)}\, dx = 2x \sin{(3x)} - \int 2\sin{(3x)}\, dx$

$\implies \int 6x\cos{(3x)}\, dx = 2x \sin{(3x)} + \frac{2}{3} \cos{(3x)} + C$

By the fundamental theorem of calculus, if $\int f(x)\, dx = F(x)$ , then $\int_a^b f(x)\, dx = F(b) - F(a)$

Therefore $\int_0^{\frac{\pi}{6}} 6x\cos{(3x)}\, dx = \left[2x \sin{(3x)} + \frac{2}{3} \cos{(3x)}\right]_0^{\frac{\pi}{6}}$

$= \frac{\pi}{3}\sin\left(\frac{\pi}{2}\right) + \frac{2}{3}\cos\left(\frac{\pi}{2}\right) - \left[0 + \frac{2}{3}\cos{(0)}\right]$

$= \frac{\pi}{3} - \frac{2}{3}$

Therefore $\int_0^{\frac{\pi}{6}} x\cos{(3x)}\, dx = \frac{1}{6}\int_0^{\frac{\pi}{6}} 6x\cos{(3x)}\, dx = \frac{\pi}{18} - \frac{1}{9}$

lauzy358 · « **Reply #16 on:** November 08, 2008, 09:12:45 pm »

Quote from: BiG DaN on November 06, 2008, 11:01:22 am

this is just antidifferentiation by recognition yeh?

yepp.
& thanks for posting, trinon

zzdfa · « **Reply #17 on:** November 12, 2008, 03:49:03 pm »

if you generalize this you get integration by parts =D

Glockmeister · « **Reply #18 on:** November 12, 2008, 04:06:47 pm »

yes we do.. i think someone mentioned it to trinon on the night he published this HOW TO guide

schnappy · « **Reply #19 on:** August 04, 2010, 03:56:18 pm »

There's a chapter in my text book on this... 'Integration by Recognition'. Is it a common thing in the exams? I don't really 'get it' too well... also seams pointless. Are the VCAA known to ever ask questions like this? Edit: Well I've worked it out after a bit more though... rather than just showing maths it really should be explained better in the OP, imo...

8039 · « **Reply #20 on:** October 05, 2010, 03:40:04 am »

Can someone explain wth is this? and why multiply by x?

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Author Topic: HOW TO: Anti derivatives through derivatives. (Read 3570 times) Tweet Share

Collin Li

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lauzy358

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zzdfa

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Glockmeister

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schnappy

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8039

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