Mathematics
OpenStudy (anonymous):

Integrate (2x+5)/(x^2+2x+2)dx

OpenStudy (anonymous):

partial fractions or u-sub?

OpenStudy (saifoo.khan):

$\Huge \frac{\cancel{new-trick}}{\cancel{new-trick}}$

OpenStudy (anonymous):

how did i guess?

OpenStudy (anonymous):

i tried partial fractions but the denomintor doesnt factorise

OpenStudy (anonymous):

Let's try substitution $$\alpha(\frac{d}{dx}(x^2 +2x+2)) + \beta = 2x+5$$

OpenStudy (anonymous):

damn. well it does if you use complex numbers and then contour, but you are right, so i guess it is a u - sub.

OpenStudy (anonymous):

so u=x^2+2x+2? or do i need to make the fraction into a different form?

OpenStudy (anonymous):

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

OpenStudy (anonymous):

i think that is what ishaan had yes?

OpenStudy (anonymous):

right, i'll work with that. thank you

OpenStudy (anonymous):

wait hold on

OpenStudy (anonymous):

maybe it should be 2x+2 instead of 2x+3?

OpenStudy (anonymous):

should be $\int \frac{2x+2}{x^2+2x+2}dx+\int\frac{3}{x^2+2x+2}dx$

OpenStudy (anonymous):

yeah i screwed up it is early

OpenStudy (anonymous):

so do you get ln(x^2+2x+2)+3ln(x^2+2x+2)+c

OpenStudy (anonymous):

Let me check

OpenStudy (anonymous):

actualy thats wrong sorry

OpenStudy (anonymous):

ln(x^2+2x+2)=-3/x+6ln(x)+6x+c

OpenStudy (anonymous):

ah $$log(2x^2 +2x +2) + 3 tan^{-1}(x+1) +C$$ is what I get

OpenStudy (anonymous):

how did u get the tan bit :(

OpenStudy (anonymous):

$x^2 + 2x + 1 + 1 = (x +1)^2 + 1$

OpenStudy (anonymous):

The second part is $$\int \frac{3}{(x+1)^2 +1}dx$$

OpenStudy (anonymous):

is there no way i could just take the quadratic up and get $\int\limits_{?}^{?}3x ^{-2}+6x ^{-1}+6 dx$

OpenStudy (anonymous):

oh no because the algebra is wrong

OpenStudy (anonymous):

$\frac{3}{100+20+1}\neq \frac{3}{100}+\frac{3}{20}+\frac{3}{1}$

OpenStudy (anonymous):

oh yeah lol now i feel dumb

OpenStudy (amistre64):

if you take the quad up; you could try IBPs

OpenStudy (anonymous):

ishaan as the right trick. any other way will be much much more complicated

OpenStudy (anonymous):

IBPs ?

OpenStudy (anonymous):

IBP use ILATE lol

OpenStudy (anonymous):

@priyab ishaan really has it. that is the best and snappiest way

OpenStudy (anonymous):

lol I didn't knew about that. thanks

OpenStudy (anonymous):

thanks guys appreciate it