Question

A little direction would be appreicated

Answer 1

How would I integrate this using the partial fraction decomposition method? \(\LARGE \int \frac{x+4}{x^2+2x+5}~dx\)

Answer 2

this equation blows my head off the gravitational force of the sun

Answer 3

aaaaand probably to the edge of the ever-expanding universe

Answer 4

Probably would be easier to rewrite the integrand.

Answer 5

You DO have to be able to factor it, in order to utilize Partial Fractions.

You may wish to break it up and simplify your life a bit.

\(\dfrac{1}{2}\int \dfrac{2x+2}{x^{2}+2x+5}\;dx + \int\dfrac{3}{x^{2}+2x+5}\;dx\)

The first piece is now a trivial substitution.
The second piece looks like it could use some Completing the Square and some trig.

Since there is no real zero to that denominator, there doesn't seem to be much of a Domain problem.

Answer 6

Yeah so \[\frac{ 2x+2 }{ x^2+2x+5 }, u = x^2+2x+5, ~ du = (2x+2)dx\] and complete the square for the other one as tkhunny mentioned then use another substitution.

Answer 7

Ok so now you have \[\frac{ 1 }{ 2 } \int\limits \frac{ 1 }{ u }du +3 \int\limits \frac{ 1 }{ x^2+2x+5 }dx\]

complete the square and you'll get

\[\frac{ lnu }{ 2}+3 \int\limits \frac{ 1 }{ (x+1)^2+4 }dx\]

now make another substitution for x+1 etc, etc, and partial fractions is crappy way to do this, but it makes sense xD.