Question

x+2/(x+1)²,Integrate the given function defined on proper domain w.r.t. x.

Answer 1

HELLO DEAR,

GIVEN:-
∫(x + 2)/(x + 1)².dx

∫(x + 1 + 1)/(x + 1)².dx

∫(x + 1)/(x + 1)².dx + ∫1/(x + 1)².dx

∫dx/(x + 1) + ∫1/(x + 1)².dx

put (x + 1) = t
=> dx = dt

∫dt/t + ∫dt/t²

log|t| - 1/t + C.

put t = (x + 1)

log|x + 1| - 1/(x + 1) + C.



I HOPE IT'S HELP YOU DEAR,
THANKS

Answer 2

Dear Student,

Solution:

∫ $\frac{x+2}{(x+1)^{2} } dx$

Split the numerator and split both terms

= ∫ $\frac{x+1+1}{(x+1)^{2} } dx$

=∫ $\frac{x+1}{(x+1)^{2} } +\frac{1}{(x+1)^{2} } dx$

= ∫$\frac{1}{x+1} dx$ + ∫$\frac{1}{(x+1)^{2} } dx$

do integration now,1st term is 1/x form and second is 1/x² ,

do directly or apply substitution

= log (x+1) - 1/(x+1) +C

is the final answer.

Hope it helps you