Question

Determine the integral

∫ [(2x+2)e ^ x2+2x+3    + x^−1 / lnx ] dx

a.    e ^ x2+2x+3  +  ln|ln x| + c  

b.   e^ x2+2x+3 + ln|ln x| 

c.    e^x2+2x+3 – 1 / 2 [ln x] ^−2

d.   e^x2+2x+3 − 1 /2 [ln x]^ −2  + c

Answer 1

$\int[(2x+2)e^{x^2+2x+3} + \cfrac{1}{xlnx}]dx=\\ =\int(2x+2)e^{x^2+2x+3}dx + \int\cfrac{1}{xlnx}dx = \\ =[t=x^2+2x+3, dt=(2x+2)dx, u = lnx, du=\cfrac{dx}{x}] =\\ =\int e^tdt + \int\cfrac{du}{u} = e^t + ln|u| + C = e^{x^2+2x+3}ln|lnx| + C$

Answer: b