stuck with integration?

Question

how do i deal with being asked to find this:
S (-x-1) / (x^2 + 1) dx

(the S is sposed to be the integration symbol. PS. whats that symbol called?

Answer 1

(-x -1) / (x^2 + 1)
You can separate this into two fractions.
= -x / (x^2 +1) - 1 / (x^2 + 1)

The first fraction can be integrated by substitution.
Let u = x^2 +1, then du = 2x dx
integral (-x / (x^2 +1)) = integral [-(1/2)(1/u) du]
= -(1/2) ln (x^2 +1)

The second fraction is just a common formula. It is arctan(x).
integral [ -1 / (x^2 +1) dx] = -arctan(x).

Answer: -(1/2) ln(x^2 +1) - arctan(x) + C

You can verify this using the link below.

Answer 2

Put x as tan(t)

then the equation becomes

Integral (-1 * (1+1tan(t))/(sec^2(t))) * (sec^2(t) dt)

The secant cancel leaving you
- 1 *Integral (1+tan(t))
Which is simply

-1 * (t + ln |sec(t)|)

Next you need to substitute the reverse of your transformation...

t = arctan(x)

so, the final answer is
-arctan(x) - 1/2 ln (x^2 +1)

PS : Just for your reference, check out this wonderful site
http://integrals.wolfram.com/index.jsp

If that site cannot integrate your function, chances are no integral of that function is known to man...!

Answer 3

I = - ∫(x + 1) / (x² + 1).dx
I = - (1/2).∫ 2x / (x² + 1).dx - ∫1 / (x² + 1).dx
I = (-1/2).log (x² + 1) - tan^(-1) x + C

Answer 4

integral