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December 21st, 2015, 07:40 AM   #1
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integrate sin2x cos4x

Why the final answer that I get is = 0? Which part of my working is wrong?
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Last edited by skipjack; December 21st, 2015 at 12:31 PM.
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December 21st, 2015, 08:10 AM   #2
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maybe someone else will decipher your hieroglyphics ... meanwhile, here is an easier method using a double angle identity

$\displaystyle \int_0^{\pi/2} \sin(2x)\cos(4x) \, dx$

$\displaystyle \int_0^{\pi/2} \sin(2x)\bigg[2\cos^2(2x)-1\bigg] \, dx$

$\displaystyle \int_0^{\pi/2} 2\sin(2x)\cos^2(2x)-\sin(2x) \, dx$

$\displaystyle \int_0^{\pi/2} 2\sin(2x)\cos^2(2x) \, dx - \int_0^{\pi/2} \sin(2x) \, dx$

first integral ...

$u = \cos(2x)$

$du=-2\sin(2x) \, dx$

$\displaystyle \int_{-1}^1 u^2 \, du - \int_0^{\pi/2} \sin(2x) \, dx$

$\bigg[\dfrac{u^3}{3} \bigg]_{-1}^1 - \bigg[-\dfrac{\cos(2x)}{2}\bigg]_0^{\pi/2}$

$\dfrac{2}{3} - 1 = -\dfrac{1}{3}$
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December 21st, 2015, 08:17 AM   #3
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When integrating by parts more than once, you must keep the same 'half' of the expression as $u$ and the same 'half' as $\mathrm d v$. If you swap, you just get back to where you were originally as you did.
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December 21st, 2015, 01:04 PM   #4
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$\displaystyle \begin{align*}\!\int_0^{\pi/2}\! \sin(2x)\cos(4x)\, dx &= \!\int_0^{\pi/2}\! \frac12\left(\sin(6x) - \sin(2x)\right)\,dx \\
&= \frac12\left[-\frac16\cos(6x) + \frac12\cos(2x)\right]_0^{\pi/2} \\
&= \frac12\left(-\frac16\cos(3\pi) + \frac12\cos(\pi) + \frac16\cos(0) -\frac12\cos(0)\right) \\
&= \frac12\left(\frac16 - \frac12 + \frac16 - \frac12\right) \\
&= -\frac13
\end{align*}$
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December 22nd, 2015, 04:46 AM   #5
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Actually you made several errors in your calculations. You start by doing an "integration by parts", taking $\displaystyle u= \sin(2x)$ and $\displaystyle dv= \cos(4x)dx$. Then $\displaystyle du= 2 \cos(2x)dx$ and $\displaystyle v= \frac{1}{4}\sin(4x)$ so that
$\displaystyle \int_0^{\pi/2} \sin(2x) \cos(4x)dx= \left[\frac{1}{4}\sin(2x)\sin(4x)\right]_0^{\pi/2}- \frac{1}{2}\int_0^{\pi/2} \cos(2x)\sin(4x)dx$. Of course, $\displaystyle \left[\frac{1}{4}\sin(2x)\sin(4x)\right]_0^{\pi/2}= 0$ so we have
$\displaystyle \int_0^{\pi/2} \sin(2x) \cos(4x)dx= - \frac{1}{2}\int_0^{\pi/2} \cos(2x)\sin(4x)dx$
That so far is correct. But now you do another integration by parts, this time choosing $\displaystyle u= \sin(4x)$, $\displaystyle dv= \cos(2x) dx$ so that $\displaystyle du= 4 \cos(4x)dx$, $\displaystyle v= \frac{1}{2}\sin(2x)$. But that just reverses the calculation you did above, integrating what you got by differentiating and differentiating what you got by integrating. It should be no surprise that this gives you, not 0, but, exactly what you started with:
$\displaystyle \int_0^{\pi/2} \sin(2x) \cos(4x)dx= - \frac{1}{2}\int_0^{\pi/2} \cos(2x)\sin(4x)dx$
$\displaystyle = -\frac{1}{2}\left[\frac{1}{2}\cos(4x)\sin(2x)\right]_0^{\pi/2}+\frac{1}{2}\left(\int_0^{\pi/2} \frac{1}{2}\sin(2x))(4 \cos(4x) dx)\right)$
(You appear to have forgotten the "-1/2" in front of the integral.)
Again, $\displaystyle \left[\frac{1}{2}\cos(4x)\sin(2x)\right]_0^{\pi/2}= 0$ so this reduces to $\displaystyle \int_0^{\pi/2} \sin(2x) \cos(4x)dx= \int_0^{\pi/2} \sin(2x) \cos(4x)dx$
Which does not say that the integral is 0, it just does not give a value for the integral.

If, doing the second integral "by parts", you chose $\displaystyle u= \cos(2x)$ and $\displaystyle dv= \cos(4x)dx$, then you would get the $\displaystyle -\frac{1}{3}$ that skeeter and skipjack give.

Last edited by skipjack; December 22nd, 2015 at 09:01 AM.
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