# Thread: Definite Intergral

1. ## Definite Intergral

$\displaystyle \int_{1}^{7} \frac{2^x}{x} dx$

I'm a having trouble starting this problem. Would you use integration by parts or something else?

Thanks in advance for any help

2. Originally Posted by reptile7383
$\displaystyle \int_{1}^{7} \frac{2^x}{x} dx$

I'm a having trouble starting this problem. Would you use integration by parts or something else?

Thanks in advance for any help
Are you expected to get an exact solution? Then I have bad news: It can't be done using a finite number of elementary functions.

An approximate solution (using technology) is your only option.

3. The study guide says to make an estimation using 3 subintervals. I really don't recall my prof. going over anything like this so I really don't know what he wants from it.

4. Originally Posted by reptile7383
The study guide says to make an estimation using 3 subintervals. I really don't recall my prof. going over anything like this so I really don't know what he wants from it.
Well it certainly helps now that you've included all of the question.

Left and Right Rectangle Rules and Midpoint Rule

Numerical Integration

Also refer to your study guide and class notes. If you've been asked to do this then it must be there in your notes somewhere.

5. Sorry for not posting the whole questions. I was focused on trying to the the code for the problem correct that I forget part of the question

Anyways thanks again for your help

6. Originally Posted by reptile7383
$\displaystyle \int_{1}^{7} \frac{2^x}{x} dx$

I'm a having trouble starting this problem. Would you use integration by parts or something else?

Thanks in advance for any help
Remember your Riemann sums? Lets use left hand.

Suppose that $\displaystyle f$ is Riemann integrable on $\displaystyle [a,b]$ then $\displaystyle \int_a^b f(x)dx\approx\sum_{k=1}^{n}f\left(\Lambda_k\right) \Delta x$

With $\displaystyle \Delta x=\frac{b-a}{n}$ and $\displaystyle \Lambda_k=a+\Delta x\cdot k$. So using $\displaystyle n=3$ gives us

$\displaystyle \Delta x=\frac{7-1}{3}=2$
$\displaystyle \Lambda_k=1+\frac{7-1}{3}k=1+2k$
$\displaystyle f\left(\Lambda_k\right)=\frac{2^{1+2k}}{1+2k}=2\fr ac{4^k}{1+2k}$

So

\displaystyle \begin{aligned}\int_1^7 \frac{2^x}{x}dx&\approx\sum_{k=1}^{3}2\cdot\frac{4 ^k}{2k+1}\cdot 2\\ &=4\sum_{k=1}^{3}\frac{4^k}{2k+1}\\ &=\frac{5744}{105}\end{aligned}