Question

what is the antiderivative of (x + 1)/(xˆ2 +2x +3) dx?

Answer 1

firstly,

notice that when we differentiate the denominator we get a multiple of the numerator

Answer 2

secondly
recall that the chain rule applied to ln(u)

\[\frac{d}{dx}\ln(u) = \frac{1}{u} \times \frac{du}{dx}\]

Answer 3

whenever you see the derivative of a f(x) multiplied by a function of f(x)

this chain rule.

in this case differentiating ln(x^2 + 2x +3) with respect to x

gets us (2x+2)/(x^2 +2x +3)

Answer 4

so we want half of that

Answer 5

*think ,not "this"

Answer 6

does that make sense?

Answer 7

simple take denominator as "t" or some othr variable differentiate n substitute.
let t =(xˆ2 +2x +3)
dt=(2x+2)dx
dt/2=(x+1)dx

so our integral becomes

integrate n resubsitute t in final answer..