$\displaystyle \int_{1}^{7} \frac{2^x}{x} dx$
I'm a having trouble starting this problem. Would you use integration by parts or something else?
Thanks in advance for any help
Well it certainly helps now that you've included all of the question.
Read this:
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.
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
$\displaystyle \begin{aligned}\int_1^7 \frac{2^x}{x}dx&\approx\sum_{k=1}^{3}2\cdot\frac{4 ^k}{2k+1}\cdot 2\\
&=4\sum_{k=1}^{3}\frac{4^k}{2k+1}\\
&=\frac{5744}{105}\end{aligned}$