Answer:
Substitution of x+1
Step-by-step explanation:
We are given that
[tex]\int \frac{1}{x^2+2x+2}dx[/tex]
[tex]\int\frac{1}{(x^2+2x+1)+1}dx[/tex]
[tex]\int\frac{1}{(x+1)^2+1^2}dx[/tex]
By using identity
[tex](a+b)^2=a^2+b^2+2ab[/tex]
Substitute x+1=t
Differentiate w.r.t x
dx=dt
Substitute the values
[tex]\int\frac{1}{t^2+1^2}dx[/tex]
[tex]\frac{1}{1}tan^{-1}\frac{t}{1}+C[/tex]
By using formula :[tex]\int\frac{1}{x^2+a^2}dx=\frac{1}{a}tan^{-1}\frac{x}{a}+C[/tex]
[tex]tan^{-1}(x+1)+C[/tex]
