intergrate (2x-3)/(x^2+16) dx

Question is intergrate by partial fractions if so what am I not seeing complete the square in the denominator? Thanks for Checking out question people.

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- Mar 3rd 2008, 05:13 PMBust2000Intergrate
intergrate (2x-3)/(x^2+16) dx

Question is intergrate by partial fractions if so what am I not seeing complete the square in the denominator? Thanks for Checking out question people. - Mar 3rd 2008, 05:16 PMJhevon
ah, i don't think partial fractions is the way to go here.

note that $\displaystyle \int \frac {2x - 3}{x^2 + 16}~dx = \int \frac {2x}{x^2 + 16}~dx - \int \frac 3{x^2 + 16}~dx$

for the first, do a substitution $\displaystyle u = x^2 + 16$

for the second, do not complete the square. instead, factor out the 16 from the denominator and then do a substitution $\displaystyle t = \frac x4$, and go for the arctangent integral - Mar 3rd 2008, 05:24 PMBust2000Question
Can you give me a little deminstration of what you are saying you are saying do a u substition of intergral uv -intergral v du. Sorry I need to work with my equation converter more it doesnt recognize intergrals still. Thanks

- Mar 3rd 2008, 05:56 PMJhevon
you are mixing this up with integration by parts, that is not what i am talking about. i am saying you must do integration by substitution. please look up this method, and try not to confuse the two

Quote:

Sorry I need to work with my equation converter more it doesnt recognize intergrals still. Thanks

now, does that integral look familiar? it should. now do a substitution $\displaystyle t = \frac x4$ to see. - Mar 3rd 2008, 06:18 PMSoroban
Hello, Bust2000!

Quote:

Intergrate: .$\displaystyle

\int\frac{2x-3}{x^2+16}\, dx $

The first integral is:.$\displaystyle \int\frac{2x\,dx}{x^2+16}$

. . Let $\displaystyle u \,=\,x^2+16\quad\Rightarrow\quad du\,=\,2x\,dx$

. . Substitute: .[$\displaystyle \int\frac{du}{u} \;=\;\ln|u| + C\quad\Rightarrow\quad \ln(x^2+16) + C$

And the second integral is of the form: .$\displaystyle \int\frac{du}{u^2+a^2}\;=\;\frac{1}{a}\arctan\left (\frac{u}{a}\right) + C$