Integral Question? HELP PLZ!?

Question

From 0 to 1, (x-1)/(x^2+3x+2)dx

Answer 1

(x-1)/(x^2+3x+2)
= (x-1)/[(x+1)(x+2)]
= 3/(x+2) - 2/(x+1)

∫3/(x+2) - 2/(x+1) dx, x from 0 to 1
= 3ln|x+2| - 2ln|x+1|, x from 0 to 1
= 3ln(3/2) - 2ln2
= 3ln3 - 5ln2
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Ideas: Do partial fraction first.
(x-1)/[(x+1)(x+2)] = A/(x+2) + B/(x+1)
Solve for A and B,
A = 3, B = -2

Answer 2

(x-1) / (x+1)(x+2) = A / (x+1) + B / (x+2)

x - 1 = A(x+2) + B(x+1)

x - 1 = (A+B)x + (2A+B)

A + B = 1

2A + B = -1

-A = 2

A = -2

B = 3

Integral -2 / (x+1) + 3 / (x+2) dx

-2(Integral 1 / (x+1) dx

-2 ln |x+1|

3 Integral 1 / (x+2) dx

3 ln |x+2|

Answer: ln [ |x+1|^-2 |x+2|^3 ]

Plugging in the number will be your work.

Answer 3

(x-1)/(x^2+3x+2)
= (x-1)/[(x+1)(x+2)
= A/(x+1) +B/(x+2)
= [A(x+2) +B(x+1)]/[(x+1)(x+2)]
= [(A+B)x + 2A+B]/[(x+1)(x+2)]
So A+B =1 , and
2A+B=-1
-A= 2 --> A= -2
Thus B = 3
So now your integral is -2/(x+1) + 3/(x+2) dx
You should be able to integrate each term using
integral du/u = ln(u)

Answer 4

quintessential of[(x+a million)/(x^2+a million)] =quintessential of [x/(x^2+a million)]+quintessential of[a million/(x^2+a million)] =I1+I2 for I1 enable x^2+a million=t then dieeerentiate it we get 2xdx=dt xdx=dt/2 I1=quintessential of (a million/2t) I1=a million/2log(t) I1=a million/2log(x^2+a million) and I2=intergal of (a million/(x^2+a million)) I2=tan inverse(x) =>ans is: a million/2log(x^2+a million)+tan inverse(x)