Evaluate using integration by parts, substitution, or both if necessary.
the intergral of cos 2x ln(sin 2x) dx
My work:
w= sin2x
dw= 2cos2xdx
1/2 dw= cos2xdx
1/2 integrsl sign ln(w)dw
u= lnw
u'= 1/w
v= w
v'=1
1/2 [(lnw)(w)- integral sign (1/w)(w) dw]
1/2 (wlnw-w)
Final Answer:
1/2 sin2xln(sin2x)-1/2 sin2x
This answer seems correct to me, but when I typed this in, the answer is said to be incorrect!
Please check my work and see if I made any mistakes.
Thank you!
Wolfram said this:
http://www.wolframalpha.com/input/?i=integral+cos+2x+ln%28sin+2x%29+dx
look at the last version of the "alternate forms"
Your steps and approach appear to be correct. However, there is a small mistake in your final answer. The correct final answer for the integral is:
1/2 sin(2x) ln|sin(2x)| - 1/4 sin^2(2x) + C
The mistake in your final answer is that you forgot to include the constant of integration (C), which is required when integrating. Make sure to always include it in your final answer.
Your approach to solving the integral is almost correct, but there is a small mistake in your calculation.
Let's go through the solution step by step:
Start by using the substitution u = sin(2x), differentiating both sides, and rearranging for dx:
du = 2cos(2x)dx
dx = du / (2cos(2x))
Now rewrite the integral using the substitution:
∫cos(2x)ln(sin(2x))dx = ∫cos(2x)ln(u) * (du / (2cos(2x)))
Notice that the cos(2x) term cancels out, so we're left with:
∫ln(u) * (du / 2)
Now, apply integration by parts. Choose u = ln(u) and dv = du/2. Then, differentiate u to get du = (1/u) * du and integrate dv to get v = (1/2)u.
The integration by parts formula is given by: ∫u * dv = uv - ∫v * du
Using this formula:
∫ln(u) * (du / 2) = (1/2)u * ln(u) - ∫(1/2)u * (1/u) * du
Simplifying the integral:
∫ln(u) * (du / 2) = (1/2)u * ln(u) - (1/2) * ∫du
∫ln(u) * (du / 2) = (1/2)u * ln(u) - (1/2)u + C
Finally, substitute back u = sin(2x):
∫cos(2x)ln(sin(2x))dx = (1/2)sin(2x)ln(sin(2x)) - (1/2)sin(2x) + C
So the correct final answer is:
(1/2)sin(2x)ln(sin(2x)) - (1/2)sin(2x) + C
Double-check your calculations and make sure you haven't missed any terms or made any sign errors.