Please integrate these for me.
x/(x^2+1)
(1-2x)/(x^2+1)
x/(x^2+1)
(1-2x)/(x^2+1)
-
In both cases you must use U substitution:
∫ x/(x²+1) dx
u=x²+1
u'=2x
2xdx=du => dx=du/2x
∫ (x/u)*(du/2x)
∫ 1/2u du => (1/2)lnu => (1/2)ln(x²+1) + C
--------------
∫ (1-2x)/(x²+1) dx
∫ 1/(x²+1) -2x/(x²+1) dx =>
arctanx -ln(x²+1) + C
∫ x/(x²+1) dx
u=x²+1
u'=2x
2xdx=du => dx=du/2x
∫ (x/u)*(du/2x)
∫ 1/2u du => (1/2)lnu => (1/2)ln(x²+1) + C
--------------
∫ (1-2x)/(x²+1) dx
∫ 1/(x²+1) -2x/(x²+1) dx =>
arctanx -ln(x²+1) + C