Definite Integral: int[2, 4] [lnx / 5x] dx

[cmnalo]

∫ lnx/5x dx interval (2,4)
u=lnx
du= d/dx(lnx)dx=(1/x)dx

1/5∫ u/xdx

(1/5)(1/2)(u^2)

1/10(lnx^2)

1/10[ln(4)^2] - 1/10[ln(2)^2] =

Where I'm I going wrong? How do I arrive at 3/10?

Answer is: 3/10(ln2)^2

 

[skeeter]

\(\displaystyle \L \int_2^4 \frac{\ln{x}}{5x} dx = \frac{1}{5} \int_2^4 \frac{\ln{x}}{x} dx\)

\(\displaystyle \L u = \ln{x}\)

\(\displaystyle \L du = \frac{1}{x} dx\)

substitute and reset limits ...

\(\displaystyle \L \frac{1}{5} \int_{\ln{2}}^{\ln{4}} u du\)

\(\displaystyle \L \frac{1}{5} \left[\frac{u^2}{2}\right]_{\ln{2}}^{\ln{4}}\)

note ... \(\displaystyle \L \ln{4} = 2\ln{2}\)

\(\displaystyle \L \frac{1}{5} \left[\frac{4\ln^2{2}}{2} - \frac{\ln^2{2}}{2}\right]\)

\(\displaystyle \L \frac{3\ln^2{2}}{10}\)

 

[cmnalo]

Skeeter-
If ln4 = 2ln2. How do you get to 4ln2^2?

 

[skeeter]

Skeeter-
If ln4 = 2ln2. How do you get to 4ln2^2?

using the FTC, you substitute ln4 for u into u²/2 ...

\(\displaystyle \L \frac{\ln^2{4}}{2} = \frac{(2\ln{2})^2}{2} = \frac{4\ln^2{2}}{2}\)