Hi,
I've been given two integrals to solve that I can't seem to figure out how to get started on. They are: $\displaystyle\int \frac{x}{\sqrt{x^2 +x + 1}} \, dx$ and $\displaystyle \int \frac{4^x +10^x}{2^x} \, dx$. Could someone help me get started?
$\displaystyle{\int}\dfrac{x}{\sqrt{x^2+x+1}}~dx =
\displaystyle{\int}\dfrac{(x-\frac 1 2)+\frac 1 2}{\sqrt{(x-\frac 1 2)^2+(\frac {\sqrt 3} 2)^2}}~dx=$
$\displaystyle{\int}\dfrac{(x-\frac 1 2)}{\sqrt{(x-\frac 1 2)^2+(\frac {\sqrt 3} 2)^2}}~dx +
\displaystyle{\int}\dfrac{\frac 1 2}{\sqrt{(x-\frac 1 2)^2+(\frac {\sqrt 3} 2)^2}}~dx$
the first integral make the sub
$u=(x-\frac 1 2)^2+(\frac {\sqrt 3} 2)^2$
the second one you can match the template in an integral table
$\displaystyle{\int}\dfrac 1 {\sqrt{x^2+a^2}}=\sinh^{-1}\left(\dfrac x a\right)$
That's creative, I like it, thanks. My only problem is that $(x-\frac{1}{2})^2 + (\frac{\sqrt{3}}{2})^2 = x^2 - x + 1$ and not $x^2 +x +1$ no?
For problems like this in the future, is there a method for finding those factors in the denominator or do you just figure it out through intuition or trial and error? I'll probably run into this stuff on a test soon so I would really appreciate knowing that.
Corrected
$\displaystyle{\int}\dfrac{x}{\sqrt{x^2+x+1}}~dx =
\displaystyle{\int}\dfrac{(x+\frac 1 2)-\frac 1 2}{\sqrt{(x+\frac 1 2)^2+(\frac {\sqrt 3} 2)^2}}~dx=$
$\displaystyle{\int}\dfrac{(x+\frac 1 2)}{\sqrt{(x+\frac 1 2)^2+(\frac {\sqrt 3} 2)^2}}~dx -
\displaystyle{\int}\dfrac{\frac 1 2}{\sqrt{(x+\frac 1 2)^2+(\frac {\sqrt 3} 2)^2}}~dx$
the first integral make the sub
$u=(x+\frac 1 2)^2+(\frac {\sqrt 3} 2)^2$
the second one you can match the template in an integral table
$\displaystyle{\int}\dfrac 1 {\sqrt{x^2+a^2}}=\sinh^{-1}\left(\dfrac x a\right)$