# integration using ln

• Dec 5th 2006, 02:56 PM
melissah
integration using ln
Studying for my math final, and I'm having some problems with some integration questions... the first:
Integrate (bottom limit of 2, upper limit of 4) (3/x)dx
this doesn't look hard, and apparently the answer is 3ln2 but I have no idea how to get there.

The second is:
Integrate (bottom limit 1, upper limit 2) [dt/(8-3t)]

anyone wanna give me a hand with this?
• Dec 5th 2006, 03:17 PM
topsquark
Quote:

Originally Posted by melissah
Studying for my math final, and I'm having some problems with some integration questions... the first:
Integrate (bottom limit of 2, upper limit of 4) (3/x)dx
this doesn't look hard, and apparently the answer is 3ln2 but I have no idea how to get there.

The second is:
Integrate (bottom limit 1, upper limit 2) [dt/(8-3t)]

anyone wanna give me a hand with this?

$\displaystyle \int_2^4 dx \frac{3}{x} = 3 \int_2^4 \frac{dx}{x}$

= $\displaystyle 3 (ln(4) - ln(2)) = 3 ln \left ( 2^2 \right ) - 3 ln(2)$

= $\displaystyle 3 \cdot 2 ln(2) - 3 ln(2) = 3 ln(2)$

================================================== ==

$\displaystyle \int_1^2 \frac{dt}{8 - 3t}$

Let $\displaystyle x = 8 - 3t$. Then $\displaystyle dx = -3 dt$ so
$\displaystyle \int_1^2 \frac{dt}{8 - 3t} = \int_5^2 dx \frac{-1}{3} \frac{1}{x}$

= $\displaystyle -\frac{1}{3}(ln(2) - ln(5)) = \frac{1}{3}(ln(5) - ln(2))$

= $\displaystyle \frac{ln(5/2)}{3}$

-Dan