Integrate following integrals.

Question

1.integral ax+b/(sqrt(ax^2+2bx+c)dx
2.integral 1+x/(1+x^2)dx
3.integral e^x+1/e^x dx

Answer 1

let z = a x^2 + 2 b x + c
then dz/dx = 2 a x + 2 b = 2 (ax+b)
so
dx = dz/2(ax+b)
then
(ax+b) dx/(sqrt(ax^2+2bx+c) = (dz/2)/z^.5
=(1/2)(z^-.5 dz)
integrate to get
= (.5)(z^.5) / .5 = z^.5
so
sqrt (a x^2 + 2 b x + c) + constant
check arithmetic I did it fast

Answer 2

(1+x) dx/(1+x^2)
= dx/(1+x^2) + x dx/(1+x^2)
tan^-1 ( x) + .5 ln(1+x^2) + constant

Answer 3

3.integral e^x+1/e^x dx
I guess maybe you mean'
3.integral (e^x+1) /e^x dx
which would be'
dx + e^-x dx
x -e^-x + constant

Answer 4

Judging by the level of problem of the 1st and 3rd, I think the second is as it was typed
find ∫ 1+x/(1+x^2)dx
= x + (1/2) ln(1+x^2) + c