# Spot the mistake.

• Jan 1st 2013, 01:18 PM
Nervous
Spot the mistake.
$\displaystyle \int \frac{sin(ln(x))cos(ln(x))}{x} dx$

$\displaystyle u = sin(ln(x))$
$\displaystyle du = \frac{cos(ln(x))}{x} dx$

$\displaystyle \int u du$

$\displaystyle \frac{1}{2} u^2 + C$

$\displaystyle \frac{1}{2}sin^2(ln(x)) + C$
• Jan 1st 2013, 01:28 PM
emakarov
Re: Spot the mistake.
There is no mistake.
• Jan 1st 2013, 01:31 PM
LordoftheFlies
Re: Spot the mistake.
There is none. I assume you are surprised because the answer book or whatever says $\displaystyle -\dfrac{1}{2}\cos^2 (\ln x)+\mathcal{C}$
Both answers only differ by a constant, which disappears once you make the integral definite:

$\displaystyle \displaystyle\int_a^b\frac{\sin (\ln x)\cos (\ln x)}{x}\;dx$

$\displaystyle =\dfrac{1}{2}(\cos^2 (\ln a)-\cos^2 (\ln b))$

$\displaystyle =\dfrac{1}{2}(\sin^2 (\ln b)-\sin^2 (\ln a))$
• Jan 1st 2013, 07:25 PM
ibdutt
Re: Spot the mistake.
Both the answers are right, You could have made the substitution as
Just change the substitution and you would get the desired answer.
u = cos(ln(x))
Now du = (-sin⁡(ln⁡(x)))/x dx
Thus the integral becomes
-∫▒udu = -u^2/2+C = - 〖cos〗^2 (ln⁡(x) )+C