Question

integrate 3x+2 / 2x – 1 dx​

Answer 1

Step-by-step explanation:

Integ.(3x+1)/(2x^2–2x+3) dx

Let 3x+1 = A(4x-2) + B

Comparing. Coef. of x and the constant term

We get A=3/4 and B=5/2. The sum becomes

Integ.{(3/4)(4x-2) +(5/2)}\(2x^2–2x+3) dx

Integ.(3/4)(4x-2)/(2x^2–2x+3).dx +integ(5/2)/[2{x^2-x+(3/2)} dx

=(3/4)log (2x^2–2x+3) + integ.( 5/4)/{(x-1/2)^2 -1/4 +3/2} dx

=(3/4) log(2x^2–2x+3). +(5/4)/{(x-1/2)^2+(√5/2)^2 dx

=(3/4)log(2x^2–2x+3)

+5/4 .tan inverse of (x-1/2)/(√5/2). +c

=(3/4) log(2x^2–2x+3) +

(5/4) tan inverse of (2x-1)/√5. + c.