Help on another Definite Integral problem

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April 12th 2009, 08:47 PM
Jim Marnell

Help on another Definite Integral problem

I'm lost on this one, just need a starting point.

$\int_{x=1}^{x=2} (6e^{3x}-\frac{1}{x})dx$

Thanks for any help!
April 12th 2009, 08:55 PM
TheEmptySet

Quote:

Originally Posted by Jim Marnell

I'm lost on this one, just need a starting point.

$\int_{x=1}^{x=2} (6e^{3x}-\frac{1}{x})dx$

Thanks for any help!

for $6e^{3x}$ use the sub u=3x

The 2nd one needs to be memorized. Here is the hint

$\frac{d}{dx}\ln(x)=\frac{1}{x}$ so

$\int \frac{1}{x}dx=...$
April 12th 2009, 08:55 PM
mollymcf2009

Quote:

Originally Posted by Jim Marnell

I'm lost on this one, just need a starting point.

$\int_{x=1}^{x=2} (6e^{3x}-\frac{1}{x})dx$

Thanks for any help!

Just integrate term by term and factor out your constants. I'll get you started:

$6 \int^{2}_{1} e^{3x} dx - \int^{2}_{1} \frac{1}{x} dx$

* For the first one, do a u-substitution with u = 3x

Does that help you get started?
April 12th 2009, 09:02 PM
Jim Marnell

yep, thanks for getting me started!
April 12th 2009, 09:15 PM
Jim Marnell

for the 1st term, would it equal, $u=3x,,,, du=3,,, then,,, 6e^u(3)?$
I think thats the term thats confusing me the most
April 12th 2009, 09:22 PM
Jhevon

Quote:

Originally Posted by Jim Marnell

for the 1st term, would it equal, $u=3x,,,, du=3,,, then,,, 6e^u(3)?$
I think thats the term thats confusing me the most

Consider $\int 6e^{3x}~dx$

Let $u = 3x$

$\Rightarrow du = 3~dx$

$\Rightarrow \frac 13 ~du = dx$

$\int 6e^{{\color{red}3x}}~{\color{blue}dx} = \int 6e^{\color{red}u} \cdot {\color{blue}\frac 13~du}$

$= 2 \int e^u~du$

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